| L(s) = 1 | + 2.30e3·2-s − 5.90e4·3-s + 3.19e6·4-s − 1.35e8·6-s − 2.10e8·7-s + 2.52e9·8-s + 3.48e9·9-s + 6.30e10·11-s − 1.88e11·12-s − 3.66e11·13-s − 4.84e11·14-s − 8.84e11·16-s + 5.08e12·17-s + 8.02e12·18-s + 7.36e12·19-s + 1.24e13·21-s + 1.44e14·22-s + 5.08e12·23-s − 1.49e14·24-s − 8.43e14·26-s − 2.05e14·27-s − 6.72e14·28-s + 1.15e15·29-s − 2.75e15·31-s − 7.33e15·32-s − 3.72e15·33-s + 1.17e16·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.52·4-s − 0.917·6-s − 0.281·7-s + 0.832·8-s + 0.333·9-s + 0.732·11-s − 0.879·12-s − 0.737·13-s − 0.447·14-s − 0.201·16-s + 0.611·17-s + 0.529·18-s + 0.275·19-s + 0.162·21-s + 1.16·22-s + 0.0255·23-s − 0.480·24-s − 1.17·26-s − 0.192·27-s − 0.429·28-s + 0.511·29-s − 0.603·31-s − 1.15·32-s − 0.422·33-s + 0.972·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.30e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 2.10e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.30e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.66e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 5.08e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 7.36e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 5.08e12T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.15e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.75e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.13e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 3.82e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.60e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 7.69e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.26e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.45e16T + 1.54e37T^{2} \) |
| 61 | \( 1 + 7.65e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 4.35e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.76e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.98e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.52e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.50e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 7.17e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.27e20T + 5.27e41T^{2} \) |
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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
−10.44523035768749061186396131617, −9.206122638588436458296447599538, −7.42385757846566231015441742399, −6.48041955358956711785907993917, −5.59715033276756379089117094266, −4.69962312363552939119804011810, −3.74203256674241779179783019709, −2.73551200462672116027122970767, −1.42424124332925028921236891201, 0,
1.42424124332925028921236891201, 2.73551200462672116027122970767, 3.74203256674241779179783019709, 4.69962312363552939119804011810, 5.59715033276756379089117094266, 6.48041955358956711785907993917, 7.42385757846566231015441742399, 9.206122638588436458296447599538, 10.44523035768749061186396131617
