| L(s) = 1 | − 1.60·2-s − 4.88·3-s − 5.43·4-s − 3.91·5-s + 7.82·6-s − 14.0·7-s + 21.5·8-s − 3.12·9-s + 6.27·10-s + 26.5·12-s + 13·13-s + 22.5·14-s + 19.1·15-s + 9.01·16-s + 3.25·17-s + 5.00·18-s − 107.·19-s + 21.3·20-s + 68.7·21-s − 105.·23-s − 105.·24-s − 109.·25-s − 20.8·26-s + 147.·27-s + 76.4·28-s − 169.·29-s − 30.6·30-s + ⋯ |
| L(s) = 1 | − 0.566·2-s − 0.940·3-s − 0.679·4-s − 0.350·5-s + 0.532·6-s − 0.759·7-s + 0.950·8-s − 0.115·9-s + 0.198·10-s + 0.638·12-s + 0.277·13-s + 0.430·14-s + 0.329·15-s + 0.140·16-s + 0.0463·17-s + 0.0655·18-s − 1.29·19-s + 0.238·20-s + 0.714·21-s − 0.954·23-s − 0.894·24-s − 0.877·25-s − 0.157·26-s + 1.04·27-s + 0.516·28-s − 1.08·29-s − 0.186·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 1.60T + 8T^{2} \) |
| 3 | \( 1 + 4.88T + 27T^{2} \) |
| 5 | \( 1 + 3.91T + 125T^{2} \) |
| 7 | \( 1 + 14.0T + 343T^{2} \) |
| 17 | \( 1 - 3.25T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 428.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 458.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 625.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 471.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 724.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 368.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 468.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 927.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 556.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 268.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.83e3T + 9.12e5T^{2} \) |
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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
−8.758402395182467752066608930421, −7.937944482087877252833659799065, −7.14520465541575251222349216617, −5.98958167388523737542677605138, −5.66753437352039765194572624441, −4.32308382856224680076303262424, −3.84175712789866980850765206884, −2.28843245603475525949535966058, −0.74959120587007466860474235366, 0,
0.74959120587007466860474235366, 2.28843245603475525949535966058, 3.84175712789866980850765206884, 4.32308382856224680076303262424, 5.66753437352039765194572624441, 5.98958167388523737542677605138, 7.14520465541575251222349216617, 7.937944482087877252833659799065, 8.758402395182467752066608930421
