| L(s) = 1 | + 2·2-s + 2·3-s + 4-s − 2·5-s + 4·6-s − 4·7-s − 2·8-s + 7·9-s − 4·10-s + 8·11-s + 2·12-s + 2·13-s − 8·14-s − 4·15-s − 4·16-s + 4·17-s + 14·18-s − 4·19-s − 2·20-s − 8·21-s + 16·22-s − 4·24-s + 25-s + 4·26-s + 22·27-s − 4·28-s + 8·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s − 0.707·8-s + 7/3·9-s − 1.26·10-s + 2.41·11-s + 0.577·12-s + 0.554·13-s − 2.13·14-s − 1.03·15-s − 16-s + 0.970·17-s + 3.29·18-s − 0.917·19-s − 0.447·20-s − 1.74·21-s + 3.41·22-s − 0.816·24-s + 1/5·25-s + 0.784·26-s + 4.23·27-s − 0.755·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.354337822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.354337822\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 18 T + 145 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 24 T^{3} - 97 T^{4} - 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 18 T^{3} + 68 T^{4} + 18 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 24 T^{3} + 223 T^{4} + 24 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 14 T^{2} - 144 T^{3} - 661 T^{4} - 144 p T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 53 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 39 T^{2} + 78 T^{3} + 4 T^{4} + 78 p T^{5} - 39 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 22 T + 267 T^{2} + 2442 T^{3} + 18628 T^{4} + 2442 p T^{5} + 267 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 96 T^{2} - 24 T^{3} + 8119 T^{4} - 24 p T^{5} - 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T - 70 T^{2} + 144 T^{3} + 2699 T^{4} + 144 p T^{5} - 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 82 T^{2} + 144 T^{3} + 4043 T^{4} + 144 p T^{5} - 82 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 102 T^{2} + 5363 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T + 30 T^{2} + 1152 T^{3} - 12461 T^{4} + 1152 p T^{5} + 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.415203042153325915714280403408, −8.159175167445902562343513687573, −7.60976537116963888834486477735, −7.47585057635906747897134028797, −7.07530387084052659853316631830, −6.98858934388703801375299692271, −6.75523878169826539666907379096, −6.45932592562941152013652533914, −6.27181354985175491471744186825, −6.06187629044709327909016780517, −5.91201945061375624048425417175, −5.12210213656408051973019799627, −5.06752791065000670970333675116, −4.53189737304267864144303863949, −4.49224624422850542773502210331, −4.13064452592503863477750881196, −4.05048485559214208465038423359, −3.57777546859040679066802749497, −3.37243343721468402506650079847, −3.20139536268897994387666769347, −3.00546221648908449843011707036, −2.41594353827370523126456088029, −1.78310007598689425396275651776, −1.35447221325611620025392050118, −0.846004990505883787091900343162,
0.846004990505883787091900343162, 1.35447221325611620025392050118, 1.78310007598689425396275651776, 2.41594353827370523126456088029, 3.00546221648908449843011707036, 3.20139536268897994387666769347, 3.37243343721468402506650079847, 3.57777546859040679066802749497, 4.05048485559214208465038423359, 4.13064452592503863477750881196, 4.49224624422850542773502210331, 4.53189737304267864144303863949, 5.06752791065000670970333675116, 5.12210213656408051973019799627, 5.91201945061375624048425417175, 6.06187629044709327909016780517, 6.27181354985175491471744186825, 6.45932592562941152013652533914, 6.75523878169826539666907379096, 6.98858934388703801375299692271, 7.07530387084052659853316631830, 7.47585057635906747897134028797, 7.60976537116963888834486477735, 8.159175167445902562343513687573, 8.415203042153325915714280403408
