| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 6·11-s − 5·13-s − 14-s + 16-s + 6·17-s + 2·19-s − 6·22-s − 5·26-s − 28-s − 9·29-s + 5·31-s + 32-s + 6·34-s + 4·37-s + 2·38-s + 3·41-s + 4·43-s − 6·44-s + 6·47-s + 49-s − 5·52-s − 9·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.80·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 1.27·22-s − 0.980·26-s − 0.188·28-s − 1.67·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s + 0.324·38-s + 0.468·41-s + 0.609·43-s − 0.904·44-s + 0.875·47-s + 1/7·49-s − 0.693·52-s − 1.23·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340323830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340323830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.65013561627925709584199958079, −7.18390331537952058369180047085, −6.10068065166347919434075606519, −5.50231629412158597509107415557, −5.08710074644457479418458152035, −4.29648092179999106868707964293, −3.29588446114273382241926729578, −2.76733702654018778655756793057, −2.05760067697970682926821059096, −0.62383747599770320223159621579,
0.62383747599770320223159621579, 2.05760067697970682926821059096, 2.76733702654018778655756793057, 3.29588446114273382241926729578, 4.29648092179999106868707964293, 5.08710074644457479418458152035, 5.50231629412158597509107415557, 6.10068065166347919434075606519, 7.18390331537952058369180047085, 7.65013561627925709584199958079
