| L(s) = 1 | + 3-s + 1.65·5-s + 1.03·7-s + 9-s − 2.22·11-s + 6.25·13-s + 1.65·15-s + 2.89·17-s + 2.96·19-s + 1.03·21-s + 4.94·23-s − 2.27·25-s + 27-s + 0.823·29-s + 6.93·31-s − 2.22·33-s + 1.70·35-s + 1.08·37-s + 6.25·39-s + 0.198·41-s − 9.90·43-s + 1.65·45-s + 8.20·47-s − 5.93·49-s + 2.89·51-s − 9.08·53-s − 3.66·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.738·5-s + 0.390·7-s + 0.333·9-s − 0.669·11-s + 1.73·13-s + 0.426·15-s + 0.701·17-s + 0.679·19-s + 0.225·21-s + 1.03·23-s − 0.455·25-s + 0.192·27-s + 0.152·29-s + 1.24·31-s − 0.386·33-s + 0.288·35-s + 0.178·37-s + 1.00·39-s + 0.0310·41-s − 1.51·43-s + 0.246·45-s + 1.19·47-s − 0.847·49-s + 0.404·51-s − 1.24·53-s − 0.494·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.584755561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584755561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 1.65T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 - 0.823T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 - 0.198T + 41T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 + 6.17T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.088457456963447559835853475654, −7.57164846635632858294351251456, −6.57345760438962204997851533616, −5.96238955412784732750332157070, −5.24223942402296750828772511914, −4.47613920515301432072758790409, −3.41086166868956514135946625677, −2.90293399395600559069537528014, −1.76696467683762281495931832928, −1.06793144911660910371492009580,
1.06793144911660910371492009580, 1.76696467683762281495931832928, 2.90293399395600559069537528014, 3.41086166868956514135946625677, 4.47613920515301432072758790409, 5.24223942402296750828772511914, 5.96238955412784732750332157070, 6.57345760438962204997851533616, 7.57164846635632858294351251456, 8.088457456963447559835853475654
