L(s) = 1 | − 3.98·2-s − 3·3-s + 7.89·4-s + 11.9·6-s + 12.5·7-s + 0.411·8-s + 9·9-s − 11·11-s − 23.6·12-s + 36.0·13-s − 50.0·14-s − 64.8·16-s + 39.7·17-s − 35.8·18-s − 148.·19-s − 37.6·21-s + 43.8·22-s − 35.0·23-s − 1.23·24-s − 143.·26-s − 27·27-s + 99.2·28-s + 88.2·29-s − 166.·31-s + 255.·32-s + 33·33-s − 158.·34-s + ⋯ |
L(s) = 1 | − 1.40·2-s − 0.577·3-s + 0.987·4-s + 0.813·6-s + 0.678·7-s + 0.0181·8-s + 0.333·9-s − 0.301·11-s − 0.569·12-s + 0.769·13-s − 0.956·14-s − 1.01·16-s + 0.566·17-s − 0.469·18-s − 1.78·19-s − 0.391·21-s + 0.425·22-s − 0.317·23-s − 0.0105·24-s − 1.08·26-s − 0.192·27-s + 0.669·28-s + 0.565·29-s − 0.966·31-s + 1.40·32-s + 0.174·33-s − 0.798·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 3.98T + 8T^{2} \) |
| 7 | \( 1 - 12.5T + 343T^{2} \) |
| 13 | \( 1 - 36.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 85.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 467.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 733.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 537.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 397.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 166.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 694.T + 9.12e5T^{2} \) |
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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
−9.446845696755070983260527581923, −8.395599738216337976149944678356, −8.081071249830831199153499838677, −6.98674031761362851065110278378, −6.14967952123068065099863185797, −5.00857649236113833338564684749, −3.98003598655552187491533392605, −2.19315995091537789856418798047, −1.19041535363754808451655650078, 0,
1.19041535363754808451655650078, 2.19315995091537789856418798047, 3.98003598655552187491533392605, 5.00857649236113833338564684749, 6.14967952123068065099863185797, 6.98674031761362851065110278378, 8.081071249830831199153499838677, 8.395599738216337976149944678356, 9.446845696755070983260527581923