| L(s) = 1 | + 4.09e3·2-s + 1.67e7·4-s + 1.10e8·5-s + 4.36e10·7-s + 6.87e10·8-s + 4.54e11·10-s + 1.07e13·11-s + 7.59e12·13-s + 1.78e14·14-s + 2.81e14·16-s − 9.83e14·17-s − 1.85e15·19-s + 1.85e15·20-s + 4.42e16·22-s + 1.35e17·23-s − 2.85e17·25-s + 3.11e16·26-s + 7.33e17·28-s − 2.01e18·29-s + 2.45e18·31-s + 1.15e18·32-s − 4.02e18·34-s + 4.84e18·35-s + 5.42e19·37-s − 7.59e18·38-s + 7.61e18·40-s − 1.26e19·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.203·5-s + 1.19·7-s + 0.353·8-s + 0.143·10-s + 1.03·11-s + 0.0904·13-s + 0.843·14-s + 0.250·16-s − 0.409·17-s − 0.192·19-s + 0.101·20-s + 0.733·22-s + 1.28·23-s − 0.958·25-s + 0.0639·26-s + 0.596·28-s − 1.05·29-s + 0.559·31-s + 0.176·32-s − 0.289·34-s + 0.242·35-s + 1.35·37-s − 0.135·38-s + 0.0717·40-s − 0.0875·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(4.650850494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.650850494\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.10e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 4.36e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.07e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 7.59e12T + 7.05e27T^{2} \) |
| 17 | \( 1 + 9.83e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.85e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.35e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.01e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 2.45e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 5.42e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.26e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.91e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 3.01e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.26e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.65e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 1.32e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.44e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.59e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 7.96e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 8.93e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.73e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 4.03e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.05e25T + 4.66e49T^{2} \) |
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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
−13.24781190451553720342464617519, −11.79553250006085865104170020161, −10.94737019665980741153385839376, −9.186330966183540817242260683218, −7.71049518162554077916053008598, −6.30743226471619605268993956242, −4.97532644209849074345700674897, −3.88190825135294270297408545370, −2.22383081921494634413821094567, −1.10530576466493310991541670925,
1.10530576466493310991541670925, 2.22383081921494634413821094567, 3.88190825135294270297408545370, 4.97532644209849074345700674897, 6.30743226471619605268993956242, 7.71049518162554077916053008598, 9.186330966183540817242260683218, 10.94737019665980741153385839376, 11.79553250006085865104170020161, 13.24781190451553720342464617519
