| L(s) = 1 | + 2.74·2-s − 7.66·3-s − 0.442·4-s + 5·5-s − 21.0·6-s + 25.2·7-s − 23.2·8-s + 31.7·9-s + 13.7·10-s + 11·11-s + 3.38·12-s − 28.9·13-s + 69.3·14-s − 38.3·15-s − 60.2·16-s − 120.·17-s + 87.2·18-s + 19·19-s − 2.21·20-s − 193.·21-s + 30.2·22-s + 89.9·23-s + 177.·24-s + 25·25-s − 79.5·26-s − 36.3·27-s − 11.1·28-s + ⋯ |
| L(s) = 1 | + 0.971·2-s − 1.47·3-s − 0.0552·4-s + 0.447·5-s − 1.43·6-s + 1.36·7-s − 1.02·8-s + 1.17·9-s + 0.434·10-s + 0.301·11-s + 0.0815·12-s − 0.617·13-s + 1.32·14-s − 0.659·15-s − 0.941·16-s − 1.71·17-s + 1.14·18-s + 0.229·19-s − 0.0247·20-s − 2.00·21-s + 0.293·22-s + 0.815·23-s + 1.51·24-s + 0.200·25-s − 0.599·26-s − 0.259·27-s − 0.0752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 2.74T + 8T^{2} \) |
| 3 | \( 1 + 7.66T + 27T^{2} \) |
| 7 | \( 1 - 25.2T + 343T^{2} \) |
| 13 | \( 1 + 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 89.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 638.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 467.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 801.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 761.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 604.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 493.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 859.T + 9.12e5T^{2} \) |
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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
−9.140980390671942206971010951862, −8.344818472255256178156206039651, −6.95870739387649667601643024955, −6.33227038598986463310560472961, −5.45033213372210107491869582649, −4.64222472075586451752515717777, −4.54635418378012657473206705427, −2.74109388659968506218136184642, −1.35660072197402426252494194916, 0,
1.35660072197402426252494194916, 2.74109388659968506218136184642, 4.54635418378012657473206705427, 4.64222472075586451752515717777, 5.45033213372210107491869582649, 6.33227038598986463310560472961, 6.95870739387649667601643024955, 8.344818472255256178156206039651, 9.140980390671942206971010951862
