| L(s) = 1 | − 3.84·2-s − 4.08·3-s + 6.74·4-s + 15.7·6-s + 27.0·7-s + 4.81·8-s − 10.2·9-s − 14.4·11-s − 27.5·12-s − 9.89·13-s − 104.·14-s − 72.4·16-s − 16.5·17-s + 39.4·18-s − 74.5·19-s − 110.·21-s + 55.4·22-s + 23·23-s − 19.6·24-s + 37.9·26-s + 152.·27-s + 182.·28-s + 202.·29-s − 8.09·31-s + 239.·32-s + 59.0·33-s + 63.5·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.786·3-s + 0.843·4-s + 1.06·6-s + 1.46·7-s + 0.212·8-s − 0.380·9-s − 0.395·11-s − 0.663·12-s − 0.211·13-s − 1.98·14-s − 1.13·16-s − 0.236·17-s + 0.517·18-s − 0.900·19-s − 1.15·21-s + 0.537·22-s + 0.208·23-s − 0.167·24-s + 0.286·26-s + 1.08·27-s + 1.23·28-s + 1.29·29-s − 0.0468·31-s + 1.32·32-s + 0.311·33-s + 0.320·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 3.84T + 8T^{2} \) |
| 3 | \( 1 + 4.08T + 27T^{2} \) |
| 7 | \( 1 - 27.0T + 343T^{2} \) |
| 11 | \( 1 + 14.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.89T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.5T + 6.85e3T^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.09T + 2.97e4T^{2} \) |
| 37 | \( 1 - 210.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 428.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 200.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 130.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 292.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 780.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 114.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−10.09269814351706533028186108303, −8.677707700834454211708229036292, −8.414215686806282994321219068698, −7.43904142022387227545624874803, −6.44237908365802348909958778674, −5.20248183891237152239623898363, −4.49240669810168794457400657672, −2.41144698408741710232158306295, −1.20984451329184593308914980226, 0,
1.20984451329184593308914980226, 2.41144698408741710232158306295, 4.49240669810168794457400657672, 5.20248183891237152239623898363, 6.44237908365802348909958778674, 7.43904142022387227545624874803, 8.414215686806282994321219068698, 8.677707700834454211708229036292, 10.09269814351706533028186108303
