| L(s) = 1 | − 2-s + 0.923·3-s + 4-s + 4.40·5-s − 0.923·6-s + 4.10·7-s − 8-s − 2.14·9-s − 4.40·10-s + 4.43·11-s + 0.923·12-s + 0.831·13-s − 4.10·14-s + 4.07·15-s + 16-s + 0.235·17-s + 2.14·18-s − 6.49·19-s + 4.40·20-s + 3.78·21-s − 4.43·22-s − 23-s − 0.923·24-s + 14.4·25-s − 0.831·26-s − 4.75·27-s + 4.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.533·3-s + 0.5·4-s + 1.97·5-s − 0.377·6-s + 1.55·7-s − 0.353·8-s − 0.715·9-s − 1.39·10-s + 1.33·11-s + 0.266·12-s + 0.230·13-s − 1.09·14-s + 1.05·15-s + 0.250·16-s + 0.0571·17-s + 0.506·18-s − 1.49·19-s + 0.985·20-s + 0.826·21-s − 0.945·22-s − 0.208·23-s − 0.188·24-s + 2.88·25-s − 0.163·26-s − 0.914·27-s + 0.775·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.322912950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.322912950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.923T + 3T^{2} \) |
| 5 | \( 1 - 4.40T + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 0.831T + 13T^{2} \) |
| 17 | \( 1 - 0.235T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 0.0614T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 - 1.38T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
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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
−8.463053311268193151701046316584, −7.49135093743480015344784989980, −6.54255845527973681705076911274, −6.11250402231366234227084865407, −5.35760707438518588275519951778, −4.57054291697946233290964043571, −3.43108986640379358627038660526, −2.19768464947197513600886848942, −1.97622796809407296952893175646, −1.13551172481454253740600206113,
1.13551172481454253740600206113, 1.97622796809407296952893175646, 2.19768464947197513600886848942, 3.43108986640379358627038660526, 4.57054291697946233290964043571, 5.35760707438518588275519951778, 6.11250402231366234227084865407, 6.54255845527973681705076911274, 7.49135093743480015344784989980, 8.463053311268193151701046316584
