| L(s) = 1 | − 3.39·3-s − 4.22·5-s − 2.00·7-s + 8.52·9-s − 4.20·11-s + 0.159·13-s + 14.3·15-s − 1.62·17-s + 4.67·19-s + 6.81·21-s − 0.367·23-s + 12.8·25-s − 18.7·27-s + 5.01·29-s − 1.37·31-s + 14.2·33-s + 8.49·35-s + 2.37·37-s − 0.541·39-s − 7.68·41-s − 8.67·43-s − 36.0·45-s + 1.90·47-s − 2.96·49-s + 5.52·51-s + 4.72·53-s + 17.7·55-s + ⋯ |
| L(s) = 1 | − 1.95·3-s − 1.89·5-s − 0.758·7-s + 2.84·9-s − 1.26·11-s + 0.0442·13-s + 3.70·15-s − 0.394·17-s + 1.07·19-s + 1.48·21-s − 0.0766·23-s + 2.57·25-s − 3.60·27-s + 0.930·29-s − 0.246·31-s + 2.48·33-s + 1.43·35-s + 0.390·37-s − 0.0866·39-s − 1.20·41-s − 1.32·43-s − 5.37·45-s + 0.278·47-s − 0.424·49-s + 0.773·51-s + 0.648·53-s + 2.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2669507547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2669507547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 101 | \( 1 + T \) |
| good | 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 0.159T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.367T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 - 2.37T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 - 1.90T + 47T^{2} \) |
| 53 | \( 1 - 4.72T + 53T^{2} \) |
| 59 | \( 1 + 1.99T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.165T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−11.38561514168956442267627210318, −10.63846896206003234582514925473, −9.863535123265183808129165641940, −8.207210775496205517479445637567, −7.27914520063735114385971017125, −6.62874493553653832863955945979, −5.34097473051533066280544314597, −4.59251612925786368258399693347, −3.42592001757907954914412036991, −0.52239323161849339045406216456,
0.52239323161849339045406216456, 3.42592001757907954914412036991, 4.59251612925786368258399693347, 5.34097473051533066280544314597, 6.62874493553653832863955945979, 7.27914520063735114385971017125, 8.207210775496205517479445637567, 9.863535123265183808129165641940, 10.63846896206003234582514925473, 11.38561514168956442267627210318
