| L(s) = 1 | − 2.21·2-s + 2.66·3-s + 2.90·4-s − 0.0303·5-s − 5.91·6-s + 2.76·7-s − 2.01·8-s + 4.11·9-s + 0.0671·10-s + 11-s + 7.76·12-s − 2.93·13-s − 6.12·14-s − 0.0808·15-s − 1.35·16-s − 2.96·17-s − 9.12·18-s − 0.00846·19-s − 0.0881·20-s + 7.38·21-s − 2.21·22-s − 2.03·23-s − 5.37·24-s − 4.99·25-s + 6.50·26-s + 2.98·27-s + 8.04·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.54·3-s + 1.45·4-s − 0.0135·5-s − 2.41·6-s + 1.04·7-s − 0.711·8-s + 1.37·9-s + 0.0212·10-s + 0.301·11-s + 2.24·12-s − 0.814·13-s − 1.63·14-s − 0.0208·15-s − 0.339·16-s − 0.719·17-s − 2.15·18-s − 0.00194·19-s − 0.0197·20-s + 1.61·21-s − 0.472·22-s − 0.424·23-s − 1.09·24-s − 0.999·25-s + 1.27·26-s + 0.573·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 0.0303T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 + 0.00846T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 + 4.55T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 8.21T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.70103309952565985257533305091, −7.22743312682568388727546415104, −6.43906979299256774073404099325, −5.25936397178085351052700781229, −4.38963464942707128481909617909, −3.68088510347088124066280101559, −2.57582995599048319302733559391, −2.00097923379763578859809243991, −1.47316207865862748596272903608, 0,
1.47316207865862748596272903608, 2.00097923379763578859809243991, 2.57582995599048319302733559391, 3.68088510347088124066280101559, 4.38963464942707128481909617909, 5.25936397178085351052700781229, 6.43906979299256774073404099325, 7.22743312682568388727546415104, 7.70103309952565985257533305091
