| L(s) = 1 | − 3·2-s − 2·3-s + 8·4-s + 5·5-s + 6·6-s − 45·8-s + 27·9-s − 15·10-s + 45·11-s − 16·12-s − 118·13-s − 10·15-s + 135·16-s − 54·17-s − 81·18-s − 121·19-s + 40·20-s − 135·22-s − 69·23-s + 90·24-s + 354·26-s − 154·27-s − 324·29-s + 30·30-s − 88·31-s − 360·32-s − 90·33-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.384·3-s + 4-s + 0.447·5-s + 0.408·6-s − 1.98·8-s + 9-s − 0.474·10-s + 1.23·11-s − 0.384·12-s − 2.51·13-s − 0.172·15-s + 2.10·16-s − 0.770·17-s − 1.06·18-s − 1.46·19-s + 0.447·20-s − 1.30·22-s − 0.625·23-s + 0.765·24-s + 2.67·26-s − 1.09·27-s − 2.07·29-s + 0.182·30-s − 0.509·31-s − 1.98·32-s − 0.474·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 45 T + 694 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 59 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 54 T - 1997 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 121 T + 7782 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 88 T - 22047 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 p T + 12 p^{2} T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 195 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T - 101798 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 597 T + 207532 T^{2} + 597 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 392 T - 73317 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 280 T - 222363 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 668 T + 57207 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 782 T + 118485 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1194 T + 720667 T^{2} + 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 902 T + p^{3} T^{2} )^{2} \) |
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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
−11.58267209736287399651128661448, −10.92346992168105237331954304574, −10.26021769626393133744860111608, −9.672811118446763062863025039888, −9.542129607324851715494398006041, −9.381113582207653616508340074642, −8.428577478688992043387629518628, −8.148237290720854196389580006949, −7.21325014567227958757266482476, −6.95935213476336409337267378419, −6.58319668320953836951187300449, −5.88957691589284154573896068572, −5.39143264305443359615586135431, −4.52165325647870607520955010196, −3.92838727740324194656104250926, −2.97775683614366429770859189630, −1.95687392609680384929042194330, −1.80984154891676207836866886978, 0, 0,
1.80984154891676207836866886978, 1.95687392609680384929042194330, 2.97775683614366429770859189630, 3.92838727740324194656104250926, 4.52165325647870607520955010196, 5.39143264305443359615586135431, 5.88957691589284154573896068572, 6.58319668320953836951187300449, 6.95935213476336409337267378419, 7.21325014567227958757266482476, 8.148237290720854196389580006949, 8.428577478688992043387629518628, 9.381113582207653616508340074642, 9.542129607324851715494398006041, 9.672811118446763062863025039888, 10.26021769626393133744860111608, 10.92346992168105237331954304574, 11.58267209736287399651128661448
