| L(s) = 1 | − 2.47·2-s + 4.14·4-s − 0.949·7-s − 5.32·8-s − 3.89·11-s − 13-s + 2.35·14-s + 4.90·16-s − 5.90·17-s + 6.35·19-s + 9.66·22-s + 3.30·23-s + 2.47·26-s − 3.94·28-s + 4.29·29-s + 7.56·31-s − 1.52·32-s + 14.6·34-s + 1.30·37-s − 15.7·38-s + 6.75·41-s − 8.29·43-s − 16.1·44-s − 8.19·46-s + 0.0128·47-s − 6.09·49-s − 4.14·52-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.07·4-s − 0.359·7-s − 1.88·8-s − 1.17·11-s − 0.277·13-s + 0.629·14-s + 1.22·16-s − 1.43·17-s + 1.45·19-s + 2.05·22-s + 0.689·23-s + 0.486·26-s − 0.744·28-s + 0.797·29-s + 1.35·31-s − 0.268·32-s + 2.51·34-s + 0.214·37-s − 2.55·38-s + 1.05·41-s − 1.26·43-s − 2.43·44-s − 1.20·46-s + 0.00187·47-s − 0.871·49-s − 0.575·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 7 | \( 1 + 0.949T + 7T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 0.0128T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 4.66T + 67T^{2} \) |
| 71 | \( 1 + 7.92T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.371987601580865230759207972106, −7.88538359120845763454127128794, −7.02458442607440694717073829364, −6.57664747036851583708711299437, −5.47018911184291963825819813687, −4.54034746236294573566897687048, −2.97639814974994469998683485539, −2.46187919199043399965103784077, −1.15172990781071902289394066853, 0,
1.15172990781071902289394066853, 2.46187919199043399965103784077, 2.97639814974994469998683485539, 4.54034746236294573566897687048, 5.47018911184291963825819813687, 6.57664747036851583708711299437, 7.02458442607440694717073829364, 7.88538359120845763454127128794, 8.371987601580865230759207972106
