| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s + 2·7-s + 2·8-s + 9-s + 2·11-s − 2·12-s + 4·13-s − 4·14-s − 4·16-s − 2·17-s − 2·18-s − 6·19-s − 4·21-s − 4·22-s + 4·23-s − 4·24-s + 8·25-s − 8·26-s + 2·27-s + 2·28-s + 20·29-s − 8·31-s + 2·32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 0.755·7-s + 0.707·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.10·13-s − 1.06·14-s − 16-s − 0.485·17-s − 0.471·18-s − 1.37·19-s − 0.872·21-s − 0.852·22-s + 0.834·23-s − 0.816·24-s + 8/5·25-s − 1.56·26-s + 0.384·27-s + 0.377·28-s + 3.71·29-s − 1.43·31-s + 0.353·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7934289178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7934289178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 17 T^{2} + 2 T^{3} + 276 T^{4} + 2 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 2 p T^{3} - 4 p T^{4} - 2 p^{2} T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 16 T^{2} + 8 p T^{3} - 39 p T^{4} + 8 p^{2} T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 12 T^{2} - 184 T^{3} - 1177 T^{4} - 184 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2 T + 25 T^{2} - 254 T^{3} - 2580 T^{4} - 254 p T^{5} + 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1422 T^{3} + 16212 T^{4} + 1422 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 101 T^{2} + 34 T^{3} + 7060 T^{4} + 34 p T^{5} - 101 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 141 T^{2} - 882 T^{3} + 7292 T^{4} - 882 p T^{5} + 141 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 12 T^{2} - 168 T^{3} + 1583 T^{4} - 168 p T^{5} + 12 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 18 T^{2} - 496 T^{3} - 6349 T^{4} - 496 p T^{5} - 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 148 T^{2} + 56 T^{3} + 17551 T^{4} + 56 p T^{5} - 148 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 100 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.059016096855535551211125899199, −7.41807386623546763877067190133, −7.39084677604276459652415096816, −7.13562873607645955257109125502, −6.87877455309099440099068956198, −6.60042115296952988465668071790, −6.31361695117274672375610122221, −6.19710880409909029833760421800, −5.93238323097838767289526698454, −5.82688353516196206132468835604, −5.21489683961224234555152167325, −5.08479353198780685771128454120, −4.75495220560467322618676249032, −4.47587673333114526764412431371, −4.46984593446111680159254729091, −4.16772240291108181670492907940, −3.68391471261339286920967007639, −3.34003485197684301604645153736, −2.90860119283647154921864352381, −2.55046676697355003831634042006, −2.34354711240764872308647232724, −1.51237059267261724464774534217, −1.42137775860113999582923988484, −0.924554048519016769314546790397, −0.53866695850370366748052411926,
0.53866695850370366748052411926, 0.924554048519016769314546790397, 1.42137775860113999582923988484, 1.51237059267261724464774534217, 2.34354711240764872308647232724, 2.55046676697355003831634042006, 2.90860119283647154921864352381, 3.34003485197684301604645153736, 3.68391471261339286920967007639, 4.16772240291108181670492907940, 4.46984593446111680159254729091, 4.47587673333114526764412431371, 4.75495220560467322618676249032, 5.08479353198780685771128454120, 5.21489683961224234555152167325, 5.82688353516196206132468835604, 5.93238323097838767289526698454, 6.19710880409909029833760421800, 6.31361695117274672375610122221, 6.60042115296952988465668071790, 6.87877455309099440099068956198, 7.13562873607645955257109125502, 7.39084677604276459652415096816, 7.41807386623546763877067190133, 8.059016096855535551211125899199
