| L(s) = 1 | − 2·5-s − 2·7-s − 3·9-s + 2·11-s + 2·13-s − 8·17-s − 4·19-s + 3·25-s + 8·29-s + 8·31-s + 4·35-s − 8·37-s + 8·41-s + 6·45-s + 6·47-s + 3·49-s − 4·53-s − 4·55-s + 12·59-s + 4·61-s + 6·63-s − 4·65-s + 4·67-s − 8·71-s − 8·73-s − 4·77-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 9-s + 0.603·11-s + 0.554·13-s − 1.94·17-s − 0.917·19-s + 3/5·25-s + 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.31·37-s + 1.24·41-s + 0.894·45-s + 0.875·47-s + 3/7·49-s − 0.549·53-s − 0.539·55-s + 1.56·59-s + 0.512·61-s + 0.755·63-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 0.936·73-s − 0.455·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80281600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80281600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 146 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 171 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} \) |
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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
−7.45667858710188735612180464320, −7.14197592834807271120609972712, −6.71057339215108948526276989507, −6.70055872351092132910810167418, −6.30790456059353384542586630642, −5.94537076206308142506292918518, −5.54161496640656778307254268134, −5.21470824247982282426105797656, −4.52257566605743223113879891819, −4.37984748533737260287526754186, −4.06138602699939504456402023129, −3.83211802077238724767489471563, −3.10797479897667339466199896997, −2.91129955101260567545079418903, −2.50476984421472546993213532171, −2.18198078908283190339760492819, −1.33058073428059915506292965665, −0.935684722776326600471101523828, 0, 0,
0.935684722776326600471101523828, 1.33058073428059915506292965665, 2.18198078908283190339760492819, 2.50476984421472546993213532171, 2.91129955101260567545079418903, 3.10797479897667339466199896997, 3.83211802077238724767489471563, 4.06138602699939504456402023129, 4.37984748533737260287526754186, 4.52257566605743223113879891819, 5.21470824247982282426105797656, 5.54161496640656778307254268134, 5.94537076206308142506292918518, 6.30790456059353384542586630642, 6.70055872351092132910810167418, 6.71057339215108948526276989507, 7.14197592834807271120609972712, 7.45667858710188735612180464320
