| L(s) = 1 | + 4.84·2-s + 15.4·4-s − 7·7-s + 36.0·8-s − 62.1·11-s − 14.0·13-s − 33.8·14-s + 50.9·16-s + 63.5·17-s + 48.7·19-s − 301.·22-s − 99.3·23-s − 68.2·26-s − 108.·28-s + 69.0·29-s − 9.68·31-s − 41.7·32-s + 307.·34-s − 240.·37-s + 235.·38-s − 335.·41-s − 51.2·43-s − 960.·44-s − 481.·46-s − 451.·47-s + 49·49-s − 217.·52-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.93·4-s − 0.377·7-s + 1.59·8-s − 1.70·11-s − 0.300·13-s − 0.646·14-s + 0.795·16-s + 0.906·17-s + 0.588·19-s − 2.91·22-s − 0.900·23-s − 0.514·26-s − 0.729·28-s + 0.442·29-s − 0.0561·31-s − 0.230·32-s + 1.55·34-s − 1.06·37-s + 1.00·38-s − 1.27·41-s − 0.181·43-s − 3.29·44-s − 1.54·46-s − 1.40·47-s + 0.142·49-s − 0.580·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.84T + 8T^{2} \) |
| 11 | \( 1 + 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.68T + 2.97e4T^{2} \) |
| 37 | \( 1 + 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 51.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 305.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 504.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 513.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.86e3T + 9.12e5T^{2} \) |
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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
−8.384331824567338309409869433142, −7.60758832949921062586527399256, −6.83522105699567632849114607417, −5.87399555005394543990128888500, −5.27513364180265903058296641855, −4.63451614880122821168500082308, −3.42287669883448047211874415962, −2.93597061187632528117779620375, −1.85440271616855420699768383103, 0,
1.85440271616855420699768383103, 2.93597061187632528117779620375, 3.42287669883448047211874415962, 4.63451614880122821168500082308, 5.27513364180265903058296641855, 5.87399555005394543990128888500, 6.83522105699567632849114607417, 7.60758832949921062586527399256, 8.384331824567338309409869433142
