| L(s) = 1 | − 0.509·2-s − 1.74·4-s − 5-s + 7-s + 1.90·8-s + 0.509·10-s + 0.761·11-s + 0.924·13-s − 0.509·14-s + 2.51·16-s + 2.41·17-s + 5.39·19-s + 1.74·20-s − 0.387·22-s + 23-s + 25-s − 0.470·26-s − 1.74·28-s + 1.88·29-s + 7.21·31-s − 5.08·32-s − 1.22·34-s − 35-s − 1.31·37-s − 2.74·38-s − 1.90·40-s − 8.00·41-s + ⋯ |
| L(s) = 1 | − 0.359·2-s − 0.870·4-s − 0.447·5-s + 0.377·7-s + 0.673·8-s + 0.160·10-s + 0.229·11-s + 0.256·13-s − 0.136·14-s + 0.628·16-s + 0.585·17-s + 1.23·19-s + 0.389·20-s − 0.0826·22-s + 0.208·23-s + 0.200·25-s − 0.0922·26-s − 0.328·28-s + 0.349·29-s + 1.29·31-s − 0.899·32-s − 0.210·34-s − 0.169·35-s − 0.216·37-s − 0.445·38-s − 0.301·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419812186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419812186\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.509T + 2T^{2} \) |
| 11 | \( 1 - 0.761T + 11T^{2} \) |
| 13 | \( 1 - 0.924T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 + 8.00T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 - 5.91T + 71T^{2} \) |
| 73 | \( 1 + 0.915T + 73T^{2} \) |
| 79 | \( 1 + 3.95T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 - 0.412T + 97T^{2} \) |
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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
−8.051187442362239913022069244948, −7.41961792799847175631439402482, −6.67388759250654225179379794500, −5.63102497800193335528206746468, −5.08489995305959079175287809612, −4.33456847798972623957180655780, −3.64896534557406659581775950631, −2.83202845067356960721666146644, −1.43749903475990619842336988150, −0.71855462920444387213485073510,
0.71855462920444387213485073510, 1.43749903475990619842336988150, 2.83202845067356960721666146644, 3.64896534557406659581775950631, 4.33456847798972623957180655780, 5.08489995305959079175287809612, 5.63102497800193335528206746468, 6.67388759250654225179379794500, 7.41961792799847175631439402482, 8.051187442362239913022069244948
