| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s + 4.47·13-s − 14-s + 16-s + 4.47·17-s − 6.47·19-s − 2·20-s + 23-s − 25-s − 4.47·26-s + 28-s + 2·29-s + 6.47·31-s − 32-s − 4.47·34-s − 2·35-s − 10.9·37-s + 6.47·38-s + 2·40-s + 6·41-s + 12.9·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s + 1.24·13-s − 0.267·14-s + 0.250·16-s + 1.08·17-s − 1.48·19-s − 0.447·20-s + 0.208·23-s − 0.200·25-s − 0.877·26-s + 0.188·28-s + 0.371·29-s + 1.16·31-s − 0.176·32-s − 0.766·34-s − 0.338·35-s − 1.79·37-s + 1.04·38-s + 0.316·40-s + 0.937·41-s + 1.97·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125489623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125489623\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.670052937494835229885227240065, −8.001045887673468619505422420169, −7.60852746896310615468825788603, −6.51603647682465973212157543356, −5.96967464427714120423892927978, −4.78464325415450556113892502830, −3.92132793546329028207118067492, −3.12997846662315613764326652849, −1.85535730111492892065131383419, −0.74008259790233754967530449671,
0.74008259790233754967530449671, 1.85535730111492892065131383419, 3.12997846662315613764326652849, 3.92132793546329028207118067492, 4.78464325415450556113892502830, 5.96967464427714120423892927978, 6.51603647682465973212157543356, 7.60852746896310615468825788603, 8.001045887673468619505422420169, 8.670052937494835229885227240065
