| L(s) = 1 | − 2-s + 3-s + 4-s + 4.21·5-s − 6-s − 4.81·7-s − 8-s + 9-s − 4.21·10-s − 4.29·11-s + 12-s + 13-s + 4.81·14-s + 4.21·15-s + 16-s − 2.64·17-s − 18-s + 0.655·19-s + 4.21·20-s − 4.81·21-s + 4.29·22-s + 1.52·23-s − 24-s + 12.7·25-s − 26-s + 27-s − 4.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.88·5-s − 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s − 1.33·10-s − 1.29·11-s + 0.288·12-s + 0.277·13-s + 1.28·14-s + 1.08·15-s + 0.250·16-s − 0.642·17-s − 0.235·18-s + 0.150·19-s + 0.943·20-s − 1.05·21-s + 0.915·22-s + 0.318·23-s − 0.204·24-s + 2.55·25-s − 0.196·26-s + 0.192·27-s − 0.909·28-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 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 0.655T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 0.992T + 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−7.43406535196505813744112861558, −6.73130532545935371133887527406, −6.24630207797053858984667690981, −5.66710244056618138891389590955, −4.85347712208228523720913277070, −3.48737393769320509789737745741, −2.76116854415718385546596756143, −2.38321975200674647942478408972, −1.36332269886466537191373427638, 0,
1.36332269886466537191373427638, 2.38321975200674647942478408972, 2.76116854415718385546596756143, 3.48737393769320509789737745741, 4.85347712208228523720913277070, 5.66710244056618138891389590955, 6.24630207797053858984667690981, 6.73130532545935371133887527406, 7.43406535196505813744112861558
