| L(s) = 1 | + 3-s + 7-s + 9-s + 5.05·11-s + 3.37·13-s + 7.18·17-s − 8.23·19-s + 21-s + 6.23·23-s + 27-s − 2·29-s + 4.62·31-s + 5.05·33-s − 4.85·37-s + 3.37·39-s − 3.37·41-s − 1.24·43-s + 49-s + 7.18·51-s + 4.62·53-s − 8.23·57-s + 11.6·59-s + 0.488·61-s + 63-s − 3.61·67-s + 6.23·69-s + 10.2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.52·11-s + 0.936·13-s + 1.74·17-s − 1.88·19-s + 0.218·21-s + 1.30·23-s + 0.192·27-s − 0.371·29-s + 0.830·31-s + 0.879·33-s − 0.798·37-s + 0.540·39-s − 0.527·41-s − 0.189·43-s + 0.142·49-s + 1.00·51-s + 0.634·53-s − 1.09·57-s + 1.51·59-s + 0.0625·61-s + 0.125·63-s − 0.441·67-s + 0.750·69-s + 1.22·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.574535681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.574535681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4.62T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.488T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 1.24T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 + 8.23T + 97T^{2} \) |
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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
−8.066197283281715542172829684900, −6.90073100734006131579281479251, −6.66788761025284282990784520556, −5.74401165260212325499734563478, −4.97764309443618997495138710730, −3.93768819095631158822209549290, −3.71960989428145252471578072829, −2.68911387066875200458298158460, −1.63246352950289539847549669438, −1.00861738215017453244413305915,
1.00861738215017453244413305915, 1.63246352950289539847549669438, 2.68911387066875200458298158460, 3.71960989428145252471578072829, 3.93768819095631158822209549290, 4.97764309443618997495138710730, 5.74401165260212325499734563478, 6.66788761025284282990784520556, 6.90073100734006131579281479251, 8.066197283281715542172829684900
