L(s) = 1 | − 4·2-s + 8·4-s − 6·7-s − 32·11-s + 38·13-s + 24·14-s − 64·16-s + 26·17-s + 100·19-s + 128·22-s − 78·23-s − 152·26-s − 48·28-s + 50·29-s − 108·31-s + 256·32-s − 104·34-s − 266·37-s − 400·38-s − 22·41-s − 442·43-s − 256·44-s + 312·46-s − 514·47-s − 307·49-s + 304·52-s + 2·53-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.323·7-s − 0.877·11-s + 0.810·13-s + 0.458·14-s − 16-s + 0.370·17-s + 1.20·19-s + 1.24·22-s − 0.707·23-s − 1.14·26-s − 0.323·28-s + 0.320·29-s − 0.625·31-s + 1.41·32-s − 0.524·34-s − 1.18·37-s − 1.70·38-s − 0.0838·41-s − 1.56·43-s − 0.877·44-s + 1.00·46-s − 1.59·47-s − 0.895·49-s + 0.810·52-s + 0.00518·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 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
−10.96926519078076475361092384039, −10.15326347085816014270893512129, −9.413637005742010778557720526341, −8.344434913247458594535211323479, −7.65281470367648967595994536620, −6.49839143912079036043922633202, −5.10283943943181689170567094516, −3.29610350142201987915407042625, −1.57425675175983214824376213776, 0,
1.57425675175983214824376213776, 3.29610350142201987915407042625, 5.10283943943181689170567094516, 6.49839143912079036043922633202, 7.65281470367648967595994536620, 8.344434913247458594535211323479, 9.413637005742010778557720526341, 10.15326347085816014270893512129, 10.96926519078076475361092384039