| L(s) = 1 | − 3·3-s + 10·5-s − 8·7-s + 9·9-s + 11·11-s + 18·13-s − 30·15-s + 46·17-s − 40·19-s + 24·21-s − 44·23-s − 25·25-s − 27·27-s + 186·29-s + 72·31-s − 33·33-s − 80·35-s − 114·37-s − 54·39-s + 174·41-s + 416·43-s + 90·45-s + 156·47-s − 279·49-s − 138·51-s − 62·53-s + 110·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.431·7-s + 1/3·9-s + 0.301·11-s + 0.384·13-s − 0.516·15-s + 0.656·17-s − 0.482·19-s + 0.249·21-s − 0.398·23-s − 1/5·25-s − 0.192·27-s + 1.19·29-s + 0.417·31-s − 0.174·33-s − 0.386·35-s − 0.506·37-s − 0.221·39-s + 0.662·41-s + 1.47·43-s + 0.298·45-s + 0.484·47-s − 0.813·49-s − 0.378·51-s − 0.160·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.862542610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862542610\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 44 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 114 T + p^{3} T^{2} \) |
| 41 | \( 1 - 174 T + p^{3} T^{2} \) |
| 43 | \( 1 - 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 156 T + p^{3} T^{2} \) |
| 53 | \( 1 + 62 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 446 T + p^{3} T^{2} \) |
| 67 | \( 1 - 956 T + p^{3} T^{2} \) |
| 71 | \( 1 - 444 T + p^{3} T^{2} \) |
| 73 | \( 1 - 306 T + p^{3} T^{2} \) |
| 79 | \( 1 - 664 T + p^{3} T^{2} \) |
| 83 | \( 1 - 124 T + p^{3} T^{2} \) |
| 89 | \( 1 - 602 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1522 T + p^{3} T^{2} \) |
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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
−10.32775118080190837051081920700, −9.745562064275421712216502882367, −8.817024503181697356010092381359, −7.67340735133012966816821250086, −6.44276940995764988881673747635, −6.01074054510561957558734155765, −4.92033500105523783403608128385, −3.68374278955887531033201926046, −2.24879595684342402732996420681, −0.884646548251175841416844724447,
0.884646548251175841416844724447, 2.24879595684342402732996420681, 3.68374278955887531033201926046, 4.92033500105523783403608128385, 6.01074054510561957558734155765, 6.44276940995764988881673747635, 7.67340735133012966816821250086, 8.817024503181697356010092381359, 9.745562064275421712216502882367, 10.32775118080190837051081920700
