L(s) = 1 | + 2·2-s + 4·4-s + 7·5-s − 15·7-s + 8·8-s + 14·10-s + 49·11-s + 14·13-s − 30·14-s + 16·16-s + 33·17-s − 19·19-s + 28·20-s + 98·22-s + 148·23-s − 76·25-s + 28·26-s − 60·28-s + 278·29-s + 94·31-s + 32·32-s + 66·34-s − 105·35-s + 160·37-s − 38·38-s + 56·40-s − 400·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.626·5-s − 0.809·7-s + 0.353·8-s + 0.442·10-s + 1.34·11-s + 0.298·13-s − 0.572·14-s + 1/4·16-s + 0.470·17-s − 0.229·19-s + 0.313·20-s + 0.949·22-s + 1.34·23-s − 0.607·25-s + 0.211·26-s − 0.404·28-s + 1.78·29-s + 0.544·31-s + 0.176·32-s + 0.332·34-s − 0.507·35-s + 0.710·37-s − 0.162·38-s + 0.221·40-s − 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.348487041\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.348487041\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + p T \) |
good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 - 49 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 33 T + p^{3} T^{2} \) |
| 23 | \( 1 - 148 T + p^{3} T^{2} \) |
| 29 | \( 1 - 278 T + p^{3} T^{2} \) |
| 31 | \( 1 - 94 T + p^{3} T^{2} \) |
| 37 | \( 1 - 160 T + p^{3} T^{2} \) |
| 41 | \( 1 + 400 T + p^{3} T^{2} \) |
| 43 | \( 1 - 73 T + p^{3} T^{2} \) |
| 47 | \( 1 + 173 T + p^{3} T^{2} \) |
| 53 | \( 1 + 170 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 419 T + p^{3} T^{2} \) |
| 67 | \( 1 - 444 T + p^{3} T^{2} \) |
| 71 | \( 1 - 952 T + p^{3} T^{2} \) |
| 73 | \( 1 + 27 T + p^{3} T^{2} \) |
| 79 | \( 1 + 556 T + p^{3} T^{2} \) |
| 83 | \( 1 - 276 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1386 T + p^{3} T^{2} \) |
| 97 | \( 1 - 130 T + p^{3} T^{2} \) |
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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
−11.27761196108891979652189913624, −10.11298159246558372357794985179, −9.425843124485034893599299697981, −8.308064909252842707794338876768, −6.73068396926731449001752891010, −6.38648167744458584255270214204, −5.16415626272052223720040167244, −3.90984578055402605358485409444, −2.83886337591772895206143345325, −1.25375087497093894010927906662,
1.25375087497093894010927906662, 2.83886337591772895206143345325, 3.90984578055402605358485409444, 5.16415626272052223720040167244, 6.38648167744458584255270214204, 6.73068396926731449001752891010, 8.308064909252842707794338876768, 9.425843124485034893599299697981, 10.11298159246558372357794985179, 11.27761196108891979652189913624