| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 9·6-s − 32·7-s − 21·8-s + 9·9-s − 12·11-s + 3·12-s + 10·13-s − 96·14-s − 71·16-s + 30·17-s + 27·18-s + 19·19-s − 96·21-s − 36·22-s + 48·23-s − 63·24-s + 30·26-s + 27·27-s − 32·28-s + 150·29-s + 224·31-s − 45·32-s − 36·33-s + 90·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.612·6-s − 1.72·7-s − 0.928·8-s + 1/3·9-s − 0.328·11-s + 0.0721·12-s + 0.213·13-s − 1.83·14-s − 1.10·16-s + 0.428·17-s + 0.353·18-s + 0.229·19-s − 0.997·21-s − 0.348·22-s + 0.435·23-s − 0.535·24-s + 0.226·26-s + 0.192·27-s − 0.215·28-s + 0.960·29-s + 1.29·31-s − 0.248·32-s − 0.189·33-s + 0.453·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.896913082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.896913082\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 196 T + p^{3} T^{2} \) |
| 47 | \( 1 - 504 T + p^{3} T^{2} \) |
| 53 | \( 1 + 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 132 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 + 740 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 122 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1050 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1006 T + p^{3} T^{2} \) |
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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
−9.162400884711309089220915521174, −8.541042836419073513659124953867, −7.34875045895627760340977891363, −6.51313367719853467537336415887, −5.87619891811380768738933573538, −4.88154620560068261153022660998, −3.88991730760092177420279424584, −3.18378966244891502821089664892, −2.58629430980809380231631523352, −0.67522657057216800636212640141,
0.67522657057216800636212640141, 2.58629430980809380231631523352, 3.18378966244891502821089664892, 3.88991730760092177420279424584, 4.88154620560068261153022660998, 5.87619891811380768738933573538, 6.51313367719853467537336415887, 7.34875045895627760340977891363, 8.541042836419073513659124953867, 9.162400884711309089220915521174
