L(s) = 1 | − 2-s + 3-s + 4-s − 2.74·5-s − 6-s + 3.39·7-s − 8-s + 9-s + 2.74·10-s − 0.860·11-s + 12-s + 13-s − 3.39·14-s − 2.74·15-s + 16-s − 4.14·17-s − 18-s − 2.70·19-s − 2.74·20-s + 3.39·21-s + 0.860·22-s + 0.937·23-s − 24-s + 2.54·25-s − 26-s + 27-s + 3.39·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.22·5-s − 0.408·6-s + 1.28·7-s − 0.353·8-s + 0.333·9-s + 0.868·10-s − 0.259·11-s + 0.288·12-s + 0.277·13-s − 0.906·14-s − 0.709·15-s + 0.250·16-s − 1.00·17-s − 0.235·18-s − 0.620·19-s − 0.614·20-s + 0.740·21-s + 0.183·22-s + 0.195·23-s − 0.204·24-s + 0.509·25-s − 0.196·26-s + 0.192·27-s + 0.641·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 + 0.860T + 11T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 0.937T + 23T^{2} \) |
| 29 | \( 1 - 0.521T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 37 | \( 1 + 0.373T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 + 9.15T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.99T + 61T^{2} \) |
| 67 | \( 1 - 7.00T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 8.63T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 + 8.14T + 97T^{2} \) |
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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
−7.78752874050129978697982478254, −7.01990844084742943404805911075, −6.43939640415112385431671173708, −5.23763253995086828613914723150, −4.50235697548725985033841230536, −3.95816094365698006633423483335, −2.97733057712756067823165408234, −2.13435392783651159413947587409, −1.24753528825188337223263338883, 0,
1.24753528825188337223263338883, 2.13435392783651159413947587409, 2.97733057712756067823165408234, 3.95816094365698006633423483335, 4.50235697548725985033841230536, 5.23763253995086828613914723150, 6.43939640415112385431671173708, 7.01990844084742943404805911075, 7.78752874050129978697982478254