L(s) = 1 | + 2-s + 4-s − 4.37·5-s + 7-s + 8-s − 4.37·10-s + 1.37·11-s + 2·13-s + 14-s + 16-s − 1.37·17-s + 5·19-s − 4.37·20-s + 1.37·22-s + 1.62·23-s + 14.1·25-s + 2·26-s + 28-s + 8.74·29-s + 2·31-s + 32-s − 1.37·34-s − 4.37·35-s + 2·37-s + 5·38-s − 4.37·40-s − 4.62·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.95·5-s + 0.377·7-s + 0.353·8-s − 1.38·10-s + 0.413·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s − 0.332·17-s + 1.14·19-s − 0.977·20-s + 0.292·22-s + 0.339·23-s + 2.82·25-s + 0.392·26-s + 0.188·28-s + 1.62·29-s + 0.359·31-s + 0.176·32-s − 0.235·34-s − 0.739·35-s + 0.328·37-s + 0.811·38-s − 0.691·40-s − 0.722·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.988451066\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.988451066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 + 8.11T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.11T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−9.989311466706173968858372992494, −8.572929098740636097092393616892, −8.228202998927323849821564884408, −7.19215216282486385928762664991, −6.68189466474888002259540400955, −5.28466611498300721165837773302, −4.46922305979512963531133136284, −3.74336597601820203487671785825, −2.91612008550503675201894902122, −1.01413063779917027532365556795,
1.01413063779917027532365556795, 2.91612008550503675201894902122, 3.74336597601820203487671785825, 4.46922305979512963531133136284, 5.28466611498300721165837773302, 6.68189466474888002259540400955, 7.19215216282486385928762664991, 8.228202998927323849821564884408, 8.572929098740636097092393616892, 9.989311466706173968858372992494