L(s) = 1 | − 2-s + 4-s − 0.539·5-s + 0.739·7-s − 8-s + 0.539·10-s + 0.996·11-s − 1.00·13-s − 0.739·14-s + 16-s − 2.13·17-s − 4.61·19-s − 0.539·20-s − 0.996·22-s + 2.91·23-s − 4.70·25-s + 1.00·26-s + 0.739·28-s − 8.59·29-s − 10.1·31-s − 32-s + 2.13·34-s − 0.398·35-s − 0.483·37-s + 4.61·38-s + 0.539·40-s + 9.71·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.241·5-s + 0.279·7-s − 0.353·8-s + 0.170·10-s + 0.300·11-s − 0.280·13-s − 0.197·14-s + 0.250·16-s − 0.518·17-s − 1.05·19-s − 0.120·20-s − 0.212·22-s + 0.608·23-s − 0.941·25-s + 0.197·26-s + 0.139·28-s − 1.59·29-s − 1.81·31-s − 0.176·32-s + 0.366·34-s − 0.0673·35-s − 0.0795·37-s + 0.748·38-s + 0.0852·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9657048153\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9657048153\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 + 0.539T + 5T^{2} \) |
| 7 | \( 1 - 0.739T + 7T^{2} \) |
| 11 | \( 1 - 0.996T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 8.59T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 0.483T + 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 8.13T + 67T^{2} \) |
| 71 | \( 1 - 0.798T + 71T^{2} \) |
| 73 | \( 1 - 8.61T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.74851898054869054280616156039, −7.33182571344502366512440204363, −6.61350670835673168952562177051, −5.82673113367474875804952671679, −5.14751490466355094444214490117, −4.10382432225641323554276788653, −3.60666549072068895012246724667, −2.33164405133430787289083632368, −1.83611481475198379321136502723, −0.52498466301860434985556894880,
0.52498466301860434985556894880, 1.83611481475198379321136502723, 2.33164405133430787289083632368, 3.60666549072068895012246724667, 4.10382432225641323554276788653, 5.14751490466355094444214490117, 5.82673113367474875804952671679, 6.61350670835673168952562177051, 7.33182571344502366512440204363, 7.74851898054869054280616156039