| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 6·11-s − 12-s − 2·13-s − 14-s + 16-s + 6·17-s + 18-s + 2·19-s + 21-s + 6·22-s − 23-s − 24-s − 2·26-s − 27-s − 28-s − 6·29-s + 8·31-s + 32-s − 6·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.218·21-s + 1.27·22-s − 0.208·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.04·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.651760485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.651760485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.39169682602949, −14.66653760929541, −14.42688551424062, −13.86697585280242, −13.25975267981468, −12.58028301167212, −12.08600487613661, −11.78132365263400, −11.38979751354048, −10.46351419133985, −10.01173163354777, −9.511457536118968, −8.879889022483192, −8.058463428227275, −7.298307650268695, −6.919222618700555, −6.282262667335664, −5.708035934300115, −5.222729475765413, −4.413806097330722, −3.777364440981788, −3.357330801211246, −2.371666544717495, −1.444837639296926, −0.7612261796660349,
0.7612261796660349, 1.444837639296926, 2.371666544717495, 3.357330801211246, 3.777364440981788, 4.413806097330722, 5.222729475765413, 5.708035934300115, 6.282262667335664, 6.919222618700555, 7.298307650268695, 8.058463428227275, 8.879889022483192, 9.511457536118968, 10.01173163354777, 10.46351419133985, 11.38979751354048, 11.78132365263400, 12.08600487613661, 12.58028301167212, 13.25975267981468, 13.86697585280242, 14.42688551424062, 14.66653760929541, 15.39169682602949
