| L(s) = 1 | − 4·5-s − 9-s + 4·11-s + 4·19-s + 11·25-s + 8·29-s + 16·31-s + 12·41-s + 4·45-s + 10·49-s − 16·55-s − 12·59-s − 16·61-s + 32·79-s + 81-s − 20·89-s − 16·95-s − 4·99-s − 24·101-s − 32·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 1.20·11-s + 0.917·19-s + 11/5·25-s + 1.48·29-s + 2.87·31-s + 1.87·41-s + 0.596·45-s + 10/7·49-s − 2.15·55-s − 1.56·59-s − 2.04·61-s + 3.60·79-s + 1/9·81-s − 2.11·89-s − 1.64·95-s − 0.402·99-s − 2.38·101-s − 3.06·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053070271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053070271\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.707586333211321473024751580976, −8.060337494395150677925463991879, −8.033881182680218098054400621512, −7.67943208807894015214553161934, −7.34560550185099987603766780412, −6.70294713057692472401758815455, −6.49957259782475828552744776684, −6.35708997827238295855888812564, −5.71636113733639858439733510087, −5.14212643440291291968258974577, −4.85388046130852054282954411551, −4.24092891808335327299573207348, −4.18509644087737671562258212406, −3.83156241698335437887845646543, −3.11230195890505400617810411706, −2.77177756187010146229161505629, −2.64016681048648585783124422938, −1.39976036753184063070182800291, −1.06207106104584252948150950973, −0.52431229862945225451880123143,
0.52431229862945225451880123143, 1.06207106104584252948150950973, 1.39976036753184063070182800291, 2.64016681048648585783124422938, 2.77177756187010146229161505629, 3.11230195890505400617810411706, 3.83156241698335437887845646543, 4.18509644087737671562258212406, 4.24092891808335327299573207348, 4.85388046130852054282954411551, 5.14212643440291291968258974577, 5.71636113733639858439733510087, 6.35708997827238295855888812564, 6.49957259782475828552744776684, 6.70294713057692472401758815455, 7.34560550185099987603766780412, 7.67943208807894015214553161934, 8.033881182680218098054400621512, 8.060337494395150677925463991879, 8.707586333211321473024751580976
