| L(s) = 1 | − 2·2-s + 3-s + 3·4-s + 5-s − 2·6-s − 7-s − 4·8-s + 9-s − 2·10-s + 3·12-s − 13-s + 2·14-s + 15-s + 5·16-s + 17-s − 2·18-s + 2·19-s + 3·20-s − 21-s + 23-s − 4·24-s + 2·26-s + 27-s − 3·28-s − 2·30-s − 31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3-s + 3·4-s + 5-s − 2·6-s − 7-s − 4·8-s + 9-s − 2·10-s + 3·12-s − 13-s + 2·14-s + 15-s + 5·16-s + 17-s − 2·18-s + 2·19-s + 3·20-s − 21-s + 23-s − 4·24-s + 2·26-s + 27-s − 3·28-s − 2·30-s − 31-s − 6·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8898383634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8898383634\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 1213 | \( 1 - T \) |
| good | 2 | \( ( 1 + T )^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.157452343565893207285045643454, −8.014152865249526296084036743552, −7.39523886587276627694245060183, −7.03397902431857830181182154488, −6.06978053687039678985653480518, −5.33028842722949879724548551859, −3.28902514906025850831063880985, −2.96260887895127099918132303533, −2.01201952992933494603677889840, −1.09519217841010913826811970708,
1.09519217841010913826811970708, 2.01201952992933494603677889840, 2.96260887895127099918132303533, 3.28902514906025850831063880985, 5.33028842722949879724548551859, 6.06978053687039678985653480518, 7.03397902431857830181182154488, 7.39523886587276627694245060183, 8.014152865249526296084036743552, 9.157452343565893207285045643454
