L(s) = 1 | − 4·2-s + 16·4-s + 233.·7-s − 64·8-s + 89.7·11-s − 209.·13-s − 934.·14-s + 256·16-s − 226.·17-s − 2.18e3·19-s − 358.·22-s − 4.29e3·23-s + 836.·26-s + 3.73e3·28-s − 3.55e3·29-s + 6.90e3·31-s − 1.02e3·32-s + 907.·34-s − 5.43e3·37-s + 8.72e3·38-s − 6.48e3·41-s − 2.19e4·43-s + 1.43e3·44-s + 1.71e4·46-s + 7.71e3·47-s + 3.78e4·49-s − 3.34e3·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.80·7-s − 0.353·8-s + 0.223·11-s − 0.343·13-s − 1.27·14-s + 0.250·16-s − 0.190·17-s − 1.38·19-s − 0.158·22-s − 1.69·23-s + 0.242·26-s + 0.901·28-s − 0.785·29-s + 1.28·31-s − 0.176·32-s + 0.134·34-s − 0.652·37-s + 0.979·38-s − 0.602·41-s − 1.81·43-s + 0.111·44-s + 1.19·46-s + 0.509·47-s + 2.24·49-s − 0.171·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 233.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 89.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 209.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 226.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.96e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.84e4T + 8.58e9T^{2} \) |
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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
−9.961897107684374835223184260867, −8.598986492867825712336086994691, −8.269197551004626213009340533446, −7.30147171637064655759119770898, −6.19076695435597970548512202407, −5.00575888515818835319304954151, −4.03904863230108983270368989017, −2.25009107965815245076399840781, −1.51924495995894136039145930498, 0,
1.51924495995894136039145930498, 2.25009107965815245076399840781, 4.03904863230108983270368989017, 5.00575888515818835319304954151, 6.19076695435597970548512202407, 7.30147171637064655759119770898, 8.269197551004626213009340533446, 8.598986492867825712336086994691, 9.961897107684374835223184260867