| L(s) = 1 | − 1.12·2-s − 0.744·4-s − 4.16·5-s + 2.81·7-s + 3.07·8-s + 4.67·10-s − 4.43·11-s + 5.33·13-s − 3.15·14-s − 1.95·16-s + 2.48·17-s + 7.56·19-s + 3.10·20-s + 4.97·22-s − 23-s + 12.3·25-s − 5.97·26-s − 2.09·28-s + 29-s − 4.85·31-s − 3.95·32-s − 2.78·34-s − 11.7·35-s − 4.54·37-s − 8.47·38-s − 12.8·40-s + 10.9·41-s + ⋯ |
| L(s) = 1 | − 0.792·2-s − 0.372·4-s − 1.86·5-s + 1.06·7-s + 1.08·8-s + 1.47·10-s − 1.33·11-s + 1.47·13-s − 0.842·14-s − 0.488·16-s + 0.602·17-s + 1.73·19-s + 0.694·20-s + 1.05·22-s − 0.208·23-s + 2.47·25-s − 1.17·26-s − 0.395·28-s + 0.185·29-s − 0.871·31-s − 0.699·32-s − 0.477·34-s − 1.98·35-s − 0.747·37-s − 1.37·38-s − 2.02·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8536667494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8536667494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 7.56T + 19T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 - 9.72T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 + 7.22T + 61T^{2} \) |
| 67 | \( 1 + 2.30T + 67T^{2} \) |
| 71 | \( 1 - 1.63T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 - 0.972T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
| 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
−7.895711190907018386383832581940, −7.61824669435119429694974343073, −7.37493396258978367223960397490, −5.76932470827529815643512022574, −5.12924700084312737705883644924, −4.34282758721491253627335927270, −3.76883670019375634798023069991, −2.89833943375336501071325124171, −1.35106916702289855899711088562, −0.63242456185937326391197094646,
0.63242456185937326391197094646, 1.35106916702289855899711088562, 2.89833943375336501071325124171, 3.76883670019375634798023069991, 4.34282758721491253627335927270, 5.12924700084312737705883644924, 5.76932470827529815643512022574, 7.37493396258978367223960397490, 7.61824669435119429694974343073, 7.895711190907018386383832581940
