| L(s) = 1 | − 2·3-s + 2·5-s − 3·7-s + 3·9-s + 5·11-s − 5·13-s − 4·15-s − 3·17-s + 19-s + 6·21-s − 7·23-s + 3·25-s − 4·27-s − 5·29-s − 5·31-s − 10·33-s − 6·35-s + 11·37-s + 10·39-s + 11·41-s + 8·43-s + 6·45-s − 13·47-s + 7·49-s + 6·51-s − 28·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.13·7-s + 9-s + 1.50·11-s − 1.38·13-s − 1.03·15-s − 0.727·17-s + 0.229·19-s + 1.30·21-s − 1.45·23-s + 3/5·25-s − 0.769·27-s − 0.928·29-s − 0.898·31-s − 1.74·33-s − 1.01·35-s + 1.80·37-s + 1.60·39-s + 1.71·41-s + 1.21·43-s + 0.894·45-s − 1.89·47-s + 49-s + 0.840·51-s − 3.84·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 67 | $C_2$ | \( 1 + 16 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.157870091232138623343464981844, −7.73784481052724786911176338015, −7.35834040630965537435261229020, −7.20224440731163073983136388821, −6.52091782196506608669138277283, −6.28897632078481148633053151097, −6.14604596856987682834053507371, −5.92951557303000726612509437555, −5.27724550851888620062420352956, −5.01352060918785277498528791384, −4.37053574242863071425623343684, −4.20721839715762275028841860970, −3.81185153191444415448616824335, −3.12390699296575716746901253154, −2.65375830816605421613950706285, −2.21544893410899892571228904533, −1.53753178818260995821080194644, −1.22948114147194695122180462955, 0, 0,
1.22948114147194695122180462955, 1.53753178818260995821080194644, 2.21544893410899892571228904533, 2.65375830816605421613950706285, 3.12390699296575716746901253154, 3.81185153191444415448616824335, 4.20721839715762275028841860970, 4.37053574242863071425623343684, 5.01352060918785277498528791384, 5.27724550851888620062420352956, 5.92951557303000726612509437555, 6.14604596856987682834053507371, 6.28897632078481148633053151097, 6.52091782196506608669138277283, 7.20224440731163073983136388821, 7.35834040630965537435261229020, 7.73784481052724786911176338015, 8.157870091232138623343464981844
