| L(s) = 1 | + 1.20·3-s − 0.300·5-s + 4.45·7-s − 1.54·9-s − 11-s − 4.45·13-s − 0.362·15-s − 0.906·17-s − 19-s + 5.37·21-s − 5.49·23-s − 4.90·25-s − 5.48·27-s − 10.0·29-s − 9.95·31-s − 1.20·33-s − 1.33·35-s + 2.59·37-s − 5.37·39-s − 0.952·41-s − 7.49·43-s + 0.463·45-s + 2.50·47-s + 12.8·49-s − 1.09·51-s + 10.4·53-s + 0.300·55-s + ⋯ |
| L(s) = 1 | + 0.696·3-s − 0.134·5-s + 1.68·7-s − 0.514·9-s − 0.301·11-s − 1.23·13-s − 0.0934·15-s − 0.219·17-s − 0.229·19-s + 1.17·21-s − 1.14·23-s − 0.981·25-s − 1.05·27-s − 1.86·29-s − 1.78·31-s − 0.210·33-s − 0.226·35-s + 0.425·37-s − 0.861·39-s − 0.148·41-s − 1.14·43-s + 0.0690·45-s + 0.365·47-s + 1.83·49-s − 0.153·51-s + 1.43·53-s + 0.0404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 + 0.300T + 5T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 + 0.906T + 17T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 9.95T + 31T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 + 0.952T + 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 - 9.14T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 - 8.60T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 + 0.355T + 97T^{2} \) |
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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
−7.989008614734766289140175881258, −7.84470736463416261956256585689, −7.08551046548638726751309199007, −5.67270318612096837865784038969, −5.29750372657621531467330234522, −4.29008206148895077563153750836, −3.57566381483189874665066181538, −2.17147555669117040830742153596, −1.98385446389458977507213489795, 0,
1.98385446389458977507213489795, 2.17147555669117040830742153596, 3.57566381483189874665066181538, 4.29008206148895077563153750836, 5.29750372657621531467330234522, 5.67270318612096837865784038969, 7.08551046548638726751309199007, 7.84470736463416261956256585689, 7.989008614734766289140175881258
