| L(s) = 1 | + 2·4-s − 3·5-s + 9-s + 6·11-s − 19-s − 6·20-s + 4·25-s − 3·29-s − 2·31-s + 2·36-s − 6·41-s + 12·44-s − 3·45-s + 2·49-s − 18·55-s − 15·59-s + 19·61-s − 8·64-s − 9·71-s − 2·76-s + 79-s − 8·81-s − 27·89-s + 3·95-s + 6·99-s + 8·100-s − 3·101-s + ⋯ |
| L(s) = 1 | + 4-s − 1.34·5-s + 1/3·9-s + 1.80·11-s − 0.229·19-s − 1.34·20-s + 4/5·25-s − 0.557·29-s − 0.359·31-s + 1/3·36-s − 0.937·41-s + 1.80·44-s − 0.447·45-s + 2/7·49-s − 2.42·55-s − 1.95·59-s + 2.43·61-s − 64-s − 1.06·71-s − 0.229·76-s + 0.112·79-s − 8/9·81-s − 2.86·89-s + 0.307·95-s + 0.603·99-s + 4/5·100-s − 0.298·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037771598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037771598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
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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
−11.58692691133839009958423248535, −11.27702269848948999971040850951, −10.75280211293488605551414163339, −9.973598780879133927112653670344, −9.312001046862536482265341500143, −8.705496902004465310127456483391, −8.119737263540234289574514989364, −7.31927275417342389051953674119, −6.97104040790960408210140243816, −6.47446141033857712132803146990, −5.61761072837646834772052306310, −4.41273927766682549439809116528, −3.96569208535790183313071922941, −3.10986503405460942944420675135, −1.69653871394278332647107205445,
1.69653871394278332647107205445, 3.10986503405460942944420675135, 3.96569208535790183313071922941, 4.41273927766682549439809116528, 5.61761072837646834772052306310, 6.47446141033857712132803146990, 6.97104040790960408210140243816, 7.31927275417342389051953674119, 8.119737263540234289574514989364, 8.705496902004465310127456483391, 9.312001046862536482265341500143, 9.973598780879133927112653670344, 10.75280211293488605551414163339, 11.27702269848948999971040850951, 11.58692691133839009958423248535
