| L(s) = 1 | + 5.65·3-s − 5.65·5-s + 5.00·9-s + 4·11-s + 5.65·13-s − 32.0·15-s − 124.·17-s + 130.·19-s − 120·23-s − 93·25-s − 124.·27-s + 218·29-s − 147.·31-s + 22.6·33-s + 130·37-s + 32.0·39-s − 147.·41-s − 332·43-s − 28.2·45-s − 124.·47-s − 704·51-s − 498·53-s − 22.6·55-s + 736·57-s − 548.·59-s + 650.·61-s − 32.0·65-s + ⋯ |
| L(s) = 1 | + 1.08·3-s − 0.505·5-s + 0.185·9-s + 0.109·11-s + 0.120·13-s − 0.550·15-s − 1.77·17-s + 1.57·19-s − 1.08·23-s − 0.743·25-s − 0.887·27-s + 1.39·29-s − 0.852·31-s + 0.119·33-s + 0.577·37-s + 0.131·39-s − 0.560·41-s − 1.17·43-s − 0.0936·45-s − 0.386·47-s − 1.93·51-s − 1.29·53-s − 0.0554·55-s + 1.71·57-s − 1.21·59-s + 1.36·61-s − 0.0610·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5.65T + 27T^{2} \) |
| 5 | \( 1 + 5.65T + 125T^{2} \) |
| 11 | \( 1 - 4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.65T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 130T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 332T + 7.95e4T^{2} \) |
| 47 | \( 1 + 124.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 498T + 1.48e5T^{2} \) |
| 59 | \( 1 + 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 650.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 156T + 3.00e5T^{2} \) |
| 71 | \( 1 + 240T + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 28.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.355650994801239396758470410340, −8.543988144103333715747193660255, −7.949154855957726889711367425198, −7.07477750049422839598967193917, −6.02546287926552874916127092301, −4.72882197441834484432716993808, −3.76281328671330360446722216497, −2.88651536262354582739765553825, −1.76807405898652761854099314510, 0,
1.76807405898652761854099314510, 2.88651536262354582739765553825, 3.76281328671330360446722216497, 4.72882197441834484432716993808, 6.02546287926552874916127092301, 7.07477750049422839598967193917, 7.949154855957726889711367425198, 8.543988144103333715747193660255, 9.355650994801239396758470410340
