| L(s) = 1 | − 6.94·3-s − 49·7-s − 194.·9-s − 294.·11-s + 546.·13-s − 185.·17-s + 1.54e3·19-s + 340.·21-s + 2.64e3·23-s + 3.03e3·27-s − 4.11e3·29-s + 5.73e3·31-s + 2.04e3·33-s − 7.83e3·37-s − 3.79e3·39-s + 1.34e4·41-s + 8.92e3·43-s − 2.48e4·47-s + 2.40e3·49-s + 1.28e3·51-s − 2.14e4·53-s − 1.06e4·57-s + 1.52e4·59-s + 1.86e4·61-s + 9.54e3·63-s − 1.87e3·67-s − 1.83e4·69-s + ⋯ |
| L(s) = 1 | − 0.445·3-s − 0.377·7-s − 0.801·9-s − 0.734·11-s + 0.896·13-s − 0.155·17-s + 0.978·19-s + 0.168·21-s + 1.04·23-s + 0.802·27-s − 0.907·29-s + 1.07·31-s + 0.327·33-s − 0.941·37-s − 0.399·39-s + 1.24·41-s + 0.736·43-s − 1.64·47-s + 0.142·49-s + 0.0692·51-s − 1.04·53-s − 0.435·57-s + 0.569·59-s + 0.640·61-s + 0.303·63-s − 0.0510·67-s − 0.464·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 6.94T + 243T^{2} \) |
| 11 | \( 1 + 294.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 546.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 185.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.87e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.78e4T + 8.58e9T^{2} \) |
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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
−9.231752353018007852603882548127, −8.436729521339333907347894220891, −7.49664415754170779772810408348, −6.46047090001530251237242158273, −5.66984134392389647014214566547, −4.88568031379322956190750454327, −3.50451237032425171966249127303, −2.66247361340629818674370843848, −1.11685848126602208661648060973, 0,
1.11685848126602208661648060973, 2.66247361340629818674370843848, 3.50451237032425171966249127303, 4.88568031379322956190750454327, 5.66984134392389647014214566547, 6.46047090001530251237242158273, 7.49664415754170779772810408348, 8.436729521339333907347894220891, 9.231752353018007852603882548127
