| L(s) = 1 | + 3.53·2-s + 4.46·4-s + 2.07·5-s − 12.4·8-s + 7.33·10-s − 49.1·11-s − 44.8·13-s − 79.7·16-s − 26.5·17-s + 77.7·19-s + 9.28·20-s − 173.·22-s − 55.7·23-s − 120.·25-s − 158.·26-s − 121.·29-s + 305.·31-s − 181.·32-s − 93.6·34-s + 77.1·37-s + 274.·38-s − 25.9·40-s − 248.·41-s − 147.·43-s − 219.·44-s − 196.·46-s − 269.·47-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.558·4-s + 0.185·5-s − 0.551·8-s + 0.231·10-s − 1.34·11-s − 0.956·13-s − 1.24·16-s − 0.378·17-s + 0.938·19-s + 0.103·20-s − 1.68·22-s − 0.505·23-s − 0.965·25-s − 1.19·26-s − 0.777·29-s + 1.77·31-s − 1.00·32-s − 0.472·34-s + 0.342·37-s + 1.17·38-s − 0.102·40-s − 0.947·41-s − 0.521·43-s − 0.753·44-s − 0.630·46-s − 0.837·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.53T + 8T^{2} \) |
| 5 | \( 1 - 2.07T + 125T^{2} \) |
| 11 | \( 1 + 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 424.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 179.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 495.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−10.24857434040480258653431088824, −9.567800984833608257204516204756, −8.270914522848226906583348084408, −7.32208397941628594517430170046, −6.13394558074209423450318710344, −5.26300228119048885105944645669, −4.55539214101935584050561444063, −3.24468819522281607746867492661, −2.27823610644499134484477891011, 0,
2.27823610644499134484477891011, 3.24468819522281607746867492661, 4.55539214101935584050561444063, 5.26300228119048885105944645669, 6.13394558074209423450318710344, 7.32208397941628594517430170046, 8.270914522848226906583348084408, 9.567800984833608257204516204756, 10.24857434040480258653431088824
