| L(s) = 1 | + 2.48·2-s − 0.309·3-s + 4.17·4-s − 5-s − 0.770·6-s + 5.41·8-s − 2.90·9-s − 2.48·10-s − 11-s − 1.29·12-s − 6.09·13-s + 0.309·15-s + 5.11·16-s − 6.51·17-s − 7.21·18-s − 1.99·19-s − 4.17·20-s − 2.48·22-s − 3.32·23-s − 1.67·24-s + 25-s − 15.1·26-s + 1.82·27-s + 2.43·29-s + 0.770·30-s + 0.329·31-s + 1.86·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.178·3-s + 2.08·4-s − 0.447·5-s − 0.314·6-s + 1.91·8-s − 0.967·9-s − 0.786·10-s − 0.301·11-s − 0.373·12-s − 1.69·13-s + 0.0800·15-s + 1.27·16-s − 1.57·17-s − 1.70·18-s − 0.456·19-s − 0.934·20-s − 0.529·22-s − 0.693·23-s − 0.342·24-s + 0.200·25-s − 2.97·26-s + 0.352·27-s + 0.451·29-s + 0.140·30-s + 0.0591·31-s + 0.330·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 3 | \( 1 + 0.309T + 3T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 - 0.329T + 31T^{2} \) |
| 37 | \( 1 - 3.90T + 37T^{2} \) |
| 41 | \( 1 - 4.32T + 41T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 9.18T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.258158603132332147356494972258, −7.38017862583773436841886455757, −6.69082732727579828649049952460, −5.98021991229368362340468506386, −5.14046420687351954197889204771, −4.58078988191924001367194039773, −3.85254074361399878979648097677, −2.65030759797813225051809826506, −2.33126691699956171924129784004, 0,
2.33126691699956171924129784004, 2.65030759797813225051809826506, 3.85254074361399878979648097677, 4.58078988191924001367194039773, 5.14046420687351954197889204771, 5.98021991229368362340468506386, 6.69082732727579828649049952460, 7.38017862583773436841886455757, 8.258158603132332147356494972258
