| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s + 2·12-s − 4·13-s − 4·14-s − 15-s − 4·16-s − 4·18-s − 2·20-s − 2·21-s + 23-s − 4·25-s − 8·26-s − 5·27-s − 4·28-s − 2·30-s − 7·31-s − 8·32-s + 2·35-s − 4·36-s − 3·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.577·12-s − 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s − 0.942·18-s − 0.447·20-s − 0.436·21-s + 0.208·23-s − 4/5·25-s − 1.56·26-s − 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s − 1.41·32-s + 0.338·35-s − 2/3·36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706867782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706867782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.72411817944717, −14.43286625755390, −13.99074369959285, −13.28550512456387, −13.07436825084878, −12.30193022302707, −12.09131140513388, −11.44801453758536, −10.97585363979125, −10.13106597041511, −9.581639547486567, −9.017328138664596, −8.560109005516345, −7.673283482017514, −7.311791986421131, −6.600660213886777, −6.030606561105828, −5.362331231769129, −4.995977329564586, −4.090884130942983, −3.657053345719502, −3.149668354511536, −2.509171665084712, −1.952631728245813, −0.3386391043862116,
0.3386391043862116, 1.952631728245813, 2.509171665084712, 3.149668354511536, 3.657053345719502, 4.090884130942983, 4.995977329564586, 5.362331231769129, 6.030606561105828, 6.600660213886777, 7.311791986421131, 7.673283482017514, 8.560109005516345, 9.017328138664596, 9.581639547486567, 10.13106597041511, 10.97585363979125, 11.44801453758536, 12.09131140513388, 12.30193022302707, 13.07436825084878, 13.28550512456387, 13.99074369959285, 14.43286625755390, 14.72411817944717
