| L(s) = 1 | + 0.214·2-s − 1.95·4-s + 1.70·5-s + 0.250·7-s − 0.849·8-s + 0.367·10-s − 3.76·11-s − 0.734·13-s + 0.0538·14-s + 3.72·16-s + 3.80·17-s + 5.09·19-s − 3.33·20-s − 0.809·22-s + 5.82·23-s − 2.08·25-s − 0.157·26-s − 0.488·28-s − 4.35·29-s − 4.65·31-s + 2.50·32-s + 0.817·34-s + 0.427·35-s + 10.8·37-s + 1.09·38-s − 1.45·40-s + 9.02·41-s + ⋯ |
| L(s) = 1 | + 0.152·2-s − 0.976·4-s + 0.763·5-s + 0.0945·7-s − 0.300·8-s + 0.116·10-s − 1.13·11-s − 0.203·13-s + 0.0143·14-s + 0.931·16-s + 0.922·17-s + 1.16·19-s − 0.745·20-s − 0.172·22-s + 1.21·23-s − 0.417·25-s − 0.0309·26-s − 0.0924·28-s − 0.807·29-s − 0.835·31-s + 0.442·32-s + 0.140·34-s + 0.0722·35-s + 1.78·37-s + 0.177·38-s − 0.229·40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569097386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569097386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 0.214T + 2T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 - 0.250T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 0.734T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.17T + 67T^{2} \) |
| 71 | \( 1 + 6.28T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 9.13T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−9.453661701249714584834494420914, −9.134939029380147230223759372436, −7.82337835862098535494096607721, −7.48822243265352581773065997083, −5.87847869829643666895347363737, −5.49490144478494257554409626556, −4.67998767658455642115481779678, −3.50694516994623051780793110711, −2.50340467914819320301926853391, −0.925404710912114247590917385911,
0.925404710912114247590917385911, 2.50340467914819320301926853391, 3.50694516994623051780793110711, 4.67998767658455642115481779678, 5.49490144478494257554409626556, 5.87847869829643666895347363737, 7.48822243265352581773065997083, 7.82337835862098535494096607721, 9.134939029380147230223759372436, 9.453661701249714584834494420914
