| L(s) = 1 | − 7.56·3-s + 15.6·5-s − 27.2·7-s + 30.2·9-s − 27.5·11-s + 14.9·13-s − 118.·15-s − 143.·19-s + 206.·21-s + 193.·23-s + 121.·25-s − 24.6·27-s + 68.0·29-s + 237.·31-s + 208.·33-s − 428.·35-s − 206.·37-s − 112.·39-s + 70.8·41-s − 335.·43-s + 474.·45-s + 380.·47-s + 401.·49-s − 163.·53-s − 432.·55-s + 1.08e3·57-s − 436.·59-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 1.40·5-s − 1.47·7-s + 1.12·9-s − 0.755·11-s + 0.318·13-s − 2.04·15-s − 1.72·19-s + 2.14·21-s + 1.75·23-s + 0.970·25-s − 0.175·27-s + 0.435·29-s + 1.37·31-s + 1.09·33-s − 2.06·35-s − 0.918·37-s − 0.463·39-s + 0.269·41-s − 1.18·43-s + 1.57·45-s + 1.17·47-s + 1.17·49-s − 0.424·53-s − 1.06·55-s + 2.51·57-s − 0.964·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 7.56T + 27T^{2} \) |
| 5 | \( 1 - 15.6T + 125T^{2} \) |
| 7 | \( 1 + 27.2T + 343T^{2} \) |
| 11 | \( 1 + 27.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 68.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 436.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 553.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 307.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 395.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 25.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 80.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−8.441281369879537509521036012602, −6.93571728723538218190791731960, −6.51751326034593247370355234083, −6.01687999271708776966913200212, −5.31034605822102828385213361825, −4.56686508233912529443701873551, −3.17394236637129419072949431344, −2.28283690745053321310317629375, −0.973897897868318435471912142239, 0,
0.973897897868318435471912142239, 2.28283690745053321310317629375, 3.17394236637129419072949431344, 4.56686508233912529443701873551, 5.31034605822102828385213361825, 6.01687999271708776966913200212, 6.51751326034593247370355234083, 6.93571728723538218190791731960, 8.441281369879537509521036012602
