| L(s) = 1 | − 1.78·2-s − 0.213·3-s + 1.17·4-s + 0.380·6-s − 3.50·7-s + 1.47·8-s − 2.95·9-s − 11-s − 0.250·12-s + 1.11·13-s + 6.25·14-s − 4.97·16-s + 3.28·17-s + 5.26·18-s + 19-s + 0.749·21-s + 1.78·22-s − 0.303·23-s − 0.314·24-s − 1.98·26-s + 1.27·27-s − 4.11·28-s − 5.23·29-s − 0.126·31-s + 5.90·32-s + 0.213·33-s − 5.84·34-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.123·3-s + 0.585·4-s + 0.155·6-s − 1.32·7-s + 0.521·8-s − 0.984·9-s − 0.301·11-s − 0.0722·12-s + 0.309·13-s + 1.67·14-s − 1.24·16-s + 0.795·17-s + 1.24·18-s + 0.229·19-s + 0.163·21-s + 0.379·22-s − 0.0633·23-s − 0.0642·24-s − 0.390·26-s + 0.244·27-s − 0.777·28-s − 0.972·29-s − 0.0228·31-s + 1.04·32-s + 0.0371·33-s − 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 + 0.213T + 3T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 23 | \( 1 + 0.303T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 0.126T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 0.293T + 41T^{2} \) |
| 43 | \( 1 + 0.180T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 5.73T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 - 0.0652T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.907757992467849624885856235133, −7.38329679175263615258083613702, −6.50102103123447261874135444957, −5.87758851616122899027903419614, −5.11230989831120979222090964622, −3.91024516131134218855572898726, −3.13710736735832182035996134358, −2.25503833281576356315239208371, −0.929033178137951673182999170680, 0,
0.929033178137951673182999170680, 2.25503833281576356315239208371, 3.13710736735832182035996134358, 3.91024516131134218855572898726, 5.11230989831120979222090964622, 5.87758851616122899027903419614, 6.50102103123447261874135444957, 7.38329679175263615258083613702, 7.907757992467849624885856235133
