| L(s) = 1 | + 4·2-s + 2·3-s + 8·4-s + 8·6-s − 23·9-s + 32·11-s + 16·12-s − 38·13-s − 64·16-s + 26·17-s − 92·18-s − 100·19-s + 128·22-s + 78·23-s − 152·26-s − 100·27-s − 50·29-s + 108·31-s − 256·32-s + 64·33-s + 104·34-s − 184·36-s − 266·37-s − 400·38-s − 76·39-s − 22·41-s − 442·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s + 4-s + 0.544·6-s − 0.851·9-s + 0.877·11-s + 0.384·12-s − 0.810·13-s − 16-s + 0.370·17-s − 1.20·18-s − 1.20·19-s + 1.24·22-s + 0.707·23-s − 1.14·26-s − 0.712·27-s − 0.320·29-s + 0.625·31-s − 1.41·32-s + 0.337·33-s + 0.524·34-s − 0.851·36-s − 1.18·37-s − 1.70·38-s − 0.312·39-s − 0.0838·41-s − 1.56·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 514 T + p^{3} T^{2} \) |
| 53 | \( 1 + 2 T + p^{3} T^{2} \) |
| 59 | \( 1 + 500 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 126 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 878 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 - 282 T + p^{3} T^{2} \) |
| 89 | \( 1 - 150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−8.887742131654996575089293281878, −8.162477297389007211227102478631, −6.91234646997756984659847251517, −6.33411754980782401853834991051, −5.34850217767977901736317400574, −4.64835741008777700973313574605, −3.64268285220326236839184857359, −2.94748180853785410463070920938, −1.89544522072685515185572394616, 0,
1.89544522072685515185572394616, 2.94748180853785410463070920938, 3.64268285220326236839184857359, 4.64835741008777700973313574605, 5.34850217767977901736317400574, 6.33411754980782401853834991051, 6.91234646997756984659847251517, 8.162477297389007211227102478631, 8.887742131654996575089293281878
