| L(s) = 1 | − 4·2-s − 29.8·3-s + 16·4-s − 21·5-s + 119.·6-s − 64·8-s + 645.·9-s + 84·10-s + 331.·11-s − 476.·12-s + 66.8·13-s + 625.·15-s + 256·16-s + 240.·17-s − 2.58e3·18-s + 441.·19-s − 336·20-s − 1.32e3·22-s − 1.07e3·23-s + 1.90e3·24-s − 2.68e3·25-s − 267.·26-s − 1.19e4·27-s + 1.79e3·29-s − 2.50e3·30-s − 5.68e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.91·3-s + 0.5·4-s − 0.375·5-s + 1.35·6-s − 0.353·8-s + 2.65·9-s + 0.265·10-s + 0.826·11-s − 0.955·12-s + 0.109·13-s + 0.718·15-s + 0.250·16-s + 0.201·17-s − 1.87·18-s + 0.280·19-s − 0.187·20-s − 0.584·22-s − 0.422·23-s + 0.675·24-s − 0.858·25-s − 0.0775·26-s − 3.16·27-s + 0.395·29-s − 0.507·30-s − 1.06·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 29.8T + 243T^{2} \) |
| 5 | \( 1 + 21T + 3.12e3T^{2} \) |
| 11 | \( 1 - 331.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 66.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 240.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 441.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.29e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.18e4T + 8.58e9T^{2} \) |
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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
−11.89185600206978821432296945455, −11.48372305436079394489112438693, −10.43221730849915136611135318478, −9.449252962483284881127063774443, −7.69772144659416432408930442390, −6.60909883041963321481261286546, −5.65159116841718164774032895079, −4.15060958427698719736597172143, −1.31787676433988483968165880538, 0,
1.31787676433988483968165880538, 4.15060958427698719736597172143, 5.65159116841718164774032895079, 6.60909883041963321481261286546, 7.69772144659416432408930442390, 9.449252962483284881127063774443, 10.43221730849915136611135318478, 11.48372305436079394489112438693, 11.89185600206978821432296945455
