L(s) = 1 | − 1.96·2-s + 2.18·3-s + 1.87·4-s − 5-s − 4.30·6-s + 1.92·7-s + 0.237·8-s + 1.77·9-s + 1.96·10-s − 11-s + 4.10·12-s + 1.43·13-s − 3.79·14-s − 2.18·15-s − 4.22·16-s − 7.20·17-s − 3.49·18-s + 4.56·19-s − 1.87·20-s + 4.20·21-s + 1.96·22-s − 0.678·23-s + 0.518·24-s + 25-s − 2.83·26-s − 2.67·27-s + 3.62·28-s + ⋯ |
L(s) = 1 | − 1.39·2-s + 1.26·3-s + 0.939·4-s − 0.447·5-s − 1.75·6-s + 0.728·7-s + 0.0838·8-s + 0.591·9-s + 0.622·10-s − 0.301·11-s + 1.18·12-s + 0.398·13-s − 1.01·14-s − 0.564·15-s − 1.05·16-s − 1.74·17-s − 0.823·18-s + 1.04·19-s − 0.420·20-s + 0.918·21-s + 0.419·22-s − 0.141·23-s + 0.105·24-s + 0.200·25-s − 0.555·26-s − 0.515·27-s + 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 - 2.18T + 3T^{2} \) |
| 7 | \( 1 - 1.92T + 7T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 7.20T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 23 | \( 1 + 0.678T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 + 0.232T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 + 3.95T + 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 + 5.04T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 6.04T + 71T^{2} \) |
| 79 | \( 1 + 0.0160T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 8.96T + 97T^{2} \) |
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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
−8.317251093507205189748467530543, −7.58700861662377374622701937110, −7.25146352563343328185726568436, −6.13029689798385548338600336760, −4.88106092375139224981793831978, −4.14508355949049264217531295045, −3.14861696502191095339796451354, −2.19650848450830146315670018407, −1.47979579538124510919821356887, 0,
1.47979579538124510919821356887, 2.19650848450830146315670018407, 3.14861696502191095339796451354, 4.14508355949049264217531295045, 4.88106092375139224981793831978, 6.13029689798385548338600336760, 7.25146352563343328185726568436, 7.58700861662377374622701937110, 8.317251093507205189748467530543