| L(s) = 1 | + 6.16·3-s + 4.02·5-s + 18.9·7-s + 10.9·9-s − 35.7·11-s − 13·13-s + 24.8·15-s − 47.3·17-s + 21.8·19-s + 116.·21-s − 161.·23-s − 108.·25-s − 98.7·27-s − 133.·29-s − 97.4·31-s − 220.·33-s + 76.2·35-s − 33.5·37-s − 80.1·39-s + 276.·41-s − 119.·43-s + 44.2·45-s − 287.·47-s + 15.6·49-s − 291.·51-s − 205.·53-s − 143.·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.360·5-s + 1.02·7-s + 0.406·9-s − 0.978·11-s − 0.277·13-s + 0.427·15-s − 0.675·17-s + 0.263·19-s + 1.21·21-s − 1.46·23-s − 0.870·25-s − 0.703·27-s − 0.854·29-s − 0.564·31-s − 1.16·33-s + 0.368·35-s − 0.149·37-s − 0.328·39-s + 1.05·41-s − 0.424·43-s + 0.146·45-s − 0.893·47-s + 0.0455·49-s − 0.801·51-s − 0.531·53-s − 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 6.16T + 27T^{2} \) |
| 5 | \( 1 - 4.02T + 125T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 + 35.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 47.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 97.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 33.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 287.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 205.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 46.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 288.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 434.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 462.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 859.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 455.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 704.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 362.T + 9.12e5T^{2} \) |
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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
−8.457144902565926547570254399148, −7.910869030531142973379741625501, −7.42179159530311755302549093073, −6.08748406294254226291082906713, −5.27348780814630166240646904620, −4.34201529667923486295567850405, −3.36976668430585308118413142684, −2.25302122535403066846007672749, −1.83072806273555896102044361915, 0,
1.83072806273555896102044361915, 2.25302122535403066846007672749, 3.36976668430585308118413142684, 4.34201529667923486295567850405, 5.27348780814630166240646904620, 6.08748406294254226291082906713, 7.42179159530311755302549093073, 7.910869030531142973379741625501, 8.457144902565926547570254399148
