L(s) = 1 | + 2-s − 0.368·3-s + 4-s + 1.13·5-s − 0.368·6-s + 7-s + 8-s − 2.86·9-s + 1.13·10-s + 5.79·11-s − 0.368·12-s + 2.85·13-s + 14-s − 0.417·15-s + 16-s − 1.34·17-s − 2.86·18-s − 0.485·19-s + 1.13·20-s − 0.368·21-s + 5.79·22-s + 4.48·23-s − 0.368·24-s − 3.71·25-s + 2.85·26-s + 2.15·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.212·3-s + 0.5·4-s + 0.507·5-s − 0.150·6-s + 0.377·7-s + 0.353·8-s − 0.954·9-s + 0.358·10-s + 1.74·11-s − 0.106·12-s + 0.792·13-s + 0.267·14-s − 0.107·15-s + 0.250·16-s − 0.326·17-s − 0.675·18-s − 0.111·19-s + 0.253·20-s − 0.0803·21-s + 1.23·22-s + 0.935·23-s − 0.0751·24-s − 0.742·25-s + 0.560·26-s + 0.415·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.880496354\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.880496354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 0.368T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 + 0.485T + 19T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 7.42T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 - 3.97T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 + 2.93T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−8.273909840970602338445561518035, −6.92238545508101393505610402892, −6.63876237819816050916577129928, −5.85381270083623309582421247698, −5.33964468888690311444427777718, −4.40095614172252195594958306930, −3.74549989890818955727379916537, −2.89342266946666011656836467189, −1.88483631150594435304333580672, −1.00313638889665885550432846737,
1.00313638889665885550432846737, 1.88483631150594435304333580672, 2.89342266946666011656836467189, 3.74549989890818955727379916537, 4.40095614172252195594958306930, 5.33964468888690311444427777718, 5.85381270083623309582421247698, 6.63876237819816050916577129928, 6.92238545508101393505610402892, 8.273909840970602338445561518035