| L(s) = 1 | − 4·2-s + 16.4·3-s + 16·4-s − 14·5-s − 65.6·6-s − 151.·7-s − 64·8-s + 26.2·9-s + 56·10-s − 329.·11-s + 262.·12-s + 13.6·13-s + 605.·14-s − 229.·15-s + 256·16-s + 623.·17-s − 104.·18-s − 793.·19-s − 224·20-s − 2.48e3·21-s + 1.31e3·22-s − 3.23e3·23-s − 1.05e3·24-s − 2.92e3·25-s − 54.4·26-s − 3.55e3·27-s − 2.42e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.250·5-s − 0.744·6-s − 1.16·7-s − 0.353·8-s + 0.107·9-s + 0.177·10-s − 0.821·11-s + 0.526·12-s + 0.0223·13-s + 0.825·14-s − 0.263·15-s + 0.250·16-s + 0.523·17-s − 0.0762·18-s − 0.504·19-s − 0.125·20-s − 1.22·21-s + 0.581·22-s − 1.27·23-s − 0.372·24-s − 0.937·25-s − 0.0158·26-s − 0.939·27-s − 0.583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 - 16.4T + 243T^{2} \) |
| 5 | \( 1 + 14T + 3.12e3T^{2} \) |
| 7 | \( 1 + 151.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 329.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 13.6T + 3.71e5T^{2} \) |
| 17 | \( 1 - 623.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 793.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 754.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 732.T + 2.86e7T^{2} \) |
| 41 | \( 1 + 2.13e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.38e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.38e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.92e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.00e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.41e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−13.12396024908439178628430279289, −11.96058895176099567483091312906, −10.39350359789315994919877478068, −9.538834683141986159469054491055, −8.422812065789780943203469806391, −7.52398940474048355789623300019, −6.00324681371719758137507066231, −3.61163673826036969769937912113, −2.38857358386980685215281518264, 0,
2.38857358386980685215281518264, 3.61163673826036969769937912113, 6.00324681371719758137507066231, 7.52398940474048355789623300019, 8.422812065789780943203469806391, 9.538834683141986159469054491055, 10.39350359789315994919877478068, 11.96058895176099567483091312906, 13.12396024908439178628430279289
