| L(s) = 1 | − 4-s + 5·9-s − 12·11-s + 16-s + 4·19-s + 6·29-s − 16·31-s − 5·36-s − 18·41-s + 12·44-s − 12·59-s − 10·61-s − 64-s − 12·71-s − 4·76-s − 4·79-s + 16·81-s − 30·89-s − 60·99-s − 30·101-s − 22·109-s − 6·116-s + 86·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s − 3.61·11-s + 1/4·16-s + 0.917·19-s + 1.11·29-s − 2.87·31-s − 5/6·36-s − 2.81·41-s + 1.80·44-s − 1.56·59-s − 1.28·61-s − 1/8·64-s − 1.42·71-s − 0.458·76-s − 0.450·79-s + 16/9·81-s − 3.17·89-s − 6.03·99-s − 2.98·101-s − 2.10·109-s − 0.557·116-s + 7.81·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.596495746678319966689387101241, −8.368101838091213956096238172155, −7.77090568660566375346849901022, −7.75970196401084028857125162275, −7.26712398118375657056885622117, −7.04999849863942220506086099363, −6.55231658877963995169320636457, −5.77978045789144607023589471909, −5.45897309365227582328132280412, −5.21074142287028842400717180637, −4.86394865906334225918541512063, −4.49876568912956998969503122906, −3.90386514363511901233331297291, −3.40001472705529716918590440443, −2.80736117778798706238243893497, −2.65676149105354880550586291193, −1.58967331109431555723055578923, −1.53756854545559649779324046382, 0, 0,
1.53756854545559649779324046382, 1.58967331109431555723055578923, 2.65676149105354880550586291193, 2.80736117778798706238243893497, 3.40001472705529716918590440443, 3.90386514363511901233331297291, 4.49876568912956998969503122906, 4.86394865906334225918541512063, 5.21074142287028842400717180637, 5.45897309365227582328132280412, 5.77978045789144607023589471909, 6.55231658877963995169320636457, 7.04999849863942220506086099363, 7.26712398118375657056885622117, 7.75970196401084028857125162275, 7.77090568660566375346849901022, 8.368101838091213956096238172155, 8.596495746678319966689387101241
