| L(s) = 1 | + 1.27·2-s + 3-s − 0.379·4-s + 1.25·5-s + 1.27·6-s + 2.55·7-s − 3.02·8-s + 9-s + 1.59·10-s − 0.263·11-s − 0.379·12-s + 2.56·13-s + 3.25·14-s + 1.25·15-s − 3.09·16-s + 0.293·17-s + 1.27·18-s − 5.31·19-s − 0.475·20-s + 2.55·21-s − 0.335·22-s + 3.68·23-s − 3.02·24-s − 3.42·25-s + 3.27·26-s + 27-s − 0.969·28-s + ⋯ |
| L(s) = 1 | + 0.900·2-s + 0.577·3-s − 0.189·4-s + 0.560·5-s + 0.519·6-s + 0.966·7-s − 1.07·8-s + 0.333·9-s + 0.505·10-s − 0.0794·11-s − 0.109·12-s + 0.712·13-s + 0.869·14-s + 0.323·15-s − 0.774·16-s + 0.0711·17-s + 0.300·18-s − 1.21·19-s − 0.106·20-s + 0.557·21-s − 0.0714·22-s + 0.768·23-s − 0.618·24-s − 0.685·25-s + 0.641·26-s + 0.192·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.376947817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.376947817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 + 0.263T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 - 0.293T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 8.22T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 - 0.269T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 + 9.99T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 + 0.0409T + 71T^{2} \) |
| 73 | \( 1 + 0.813T + 73T^{2} \) |
| 79 | \( 1 - 0.783T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−11.82715325080314706885068413947, −10.91054156691131218312674928837, −9.729937170273521113155675921894, −8.782810481421660945442646722258, −8.055522472686312301735194422914, −6.57662475998715544081270846091, −5.50712669291323979469088018696, −4.54364325017417992965619053669, −3.46942363290858134813069061196, −1.95706458851829920094335146377,
1.95706458851829920094335146377, 3.46942363290858134813069061196, 4.54364325017417992965619053669, 5.50712669291323979469088018696, 6.57662475998715544081270846091, 8.055522472686312301735194422914, 8.782810481421660945442646722258, 9.729937170273521113155675921894, 10.91054156691131218312674928837, 11.82715325080314706885068413947
