| L(s) = 1 | + 1.44·3-s + 2.99·5-s + 1.69·7-s − 0.900·9-s − 5.69·11-s − 6.47·13-s + 4.34·15-s − 5.94·17-s − 0.280·19-s + 2.45·21-s + 3.98·25-s − 5.65·27-s + 1.20·29-s − 2.99·31-s − 8.25·33-s + 5.07·35-s − 0.265·37-s − 9.37·39-s + 5.81·41-s + 5.66·43-s − 2.69·45-s − 1.51·47-s − 4.13·49-s − 8.61·51-s − 6.60·53-s − 17.0·55-s − 0.405·57-s + ⋯ |
| L(s) = 1 | + 0.836·3-s + 1.34·5-s + 0.639·7-s − 0.300·9-s − 1.71·11-s − 1.79·13-s + 1.12·15-s − 1.44·17-s − 0.0642·19-s + 0.535·21-s + 0.796·25-s − 1.08·27-s + 0.223·29-s − 0.537·31-s − 1.43·33-s + 0.857·35-s − 0.0436·37-s − 1.50·39-s + 0.908·41-s + 0.863·43-s − 0.402·45-s − 0.221·47-s − 0.590·49-s − 1.20·51-s − 0.907·53-s − 2.30·55-s − 0.0537·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 + 0.280T + 19T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 + 2.99T + 31T^{2} \) |
| 37 | \( 1 + 0.265T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 - 3.11T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.653T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.959578477420311860733405430216, −7.56340111949914233842585062876, −6.59958329235713808508384963035, −5.63605138260329365820653667607, −5.11202895973686630051468636979, −4.42823135410550021172457478936, −2.92439018380600949393542586787, −2.38883141823854582031951971684, −1.95483726278169801592174862894, 0,
1.95483726278169801592174862894, 2.38883141823854582031951971684, 2.92439018380600949393542586787, 4.42823135410550021172457478936, 5.11202895973686630051468636979, 5.63605138260329365820653667607, 6.59958329235713808508384963035, 7.56340111949914233842585062876, 7.959578477420311860733405430216
