| L(s) = 1 | − 35.9·2-s + 230.·3-s + 780.·4-s + 434.·5-s − 8.29e3·6-s + 2.40e3·7-s − 9.63e3·8-s + 3.35e4·9-s − 1.56e4·10-s − 3.71e4·11-s + 1.80e5·12-s + 2.85e4·13-s − 8.63e4·14-s + 1.00e5·15-s − 5.29e4·16-s − 5.46e5·17-s − 1.20e6·18-s − 9.02e5·19-s + 3.38e5·20-s + 5.53e5·21-s + 1.33e6·22-s − 1.65e6·23-s − 2.22e6·24-s − 1.76e6·25-s − 1.02e6·26-s + 3.19e6·27-s + 1.87e6·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.64·3-s + 1.52·4-s + 0.310·5-s − 2.61·6-s + 0.377·7-s − 0.832·8-s + 1.70·9-s − 0.493·10-s − 0.764·11-s + 2.50·12-s + 0.277·13-s − 0.600·14-s + 0.510·15-s − 0.201·16-s − 1.58·17-s − 2.70·18-s − 1.58·19-s + 0.473·20-s + 0.621·21-s + 1.21·22-s − 1.23·23-s − 1.36·24-s − 0.903·25-s − 0.440·26-s + 1.15·27-s + 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 35.9T + 512T^{2} \) |
| 3 | \( 1 - 230.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 434.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.71e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.46e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.65e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.15e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.15e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.55e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.64e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.32e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.58e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.52e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.15e5T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.84e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.87e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.27e8T + 7.60e17T^{2} \) |
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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
−11.19188951692786461005584981953, −10.16636709153216365720692650583, −9.300843090785651537224493360921, −8.350043191944967970539373649116, −7.944647661063523005592762368491, −6.56628734094249382811141880427, −4.20590087945799695810122256995, −2.37335264983263087815734409217, −1.86403527468400059545680605573, 0,
1.86403527468400059545680605573, 2.37335264983263087815734409217, 4.20590087945799695810122256995, 6.56628734094249382811141880427, 7.944647661063523005592762368491, 8.350043191944967970539373649116, 9.300843090785651537224493360921, 10.16636709153216365720692650583, 11.19188951692786461005584981953
