L(s) = 1 | − 1.17·2-s + 0.381·4-s + 0.618·7-s + 0.726·8-s + 1.61·13-s − 0.726·14-s − 1.23·16-s − 1.90·17-s − 1.61·19-s + 1.17·23-s + 25-s − 1.90·26-s + 0.236·28-s + 1.90·29-s + 0.726·32-s + 2.23·34-s + 1.90·38-s + 1.17·41-s − 1.38·46-s − 0.618·49-s − 1.17·50-s + 0.618·52-s + 0.449·56-s − 2.23·58-s + 1.90·59-s − 0.618·61-s + 0.381·64-s + ⋯ |
L(s) = 1 | − 1.17·2-s + 0.381·4-s + 0.618·7-s + 0.726·8-s + 1.61·13-s − 0.726·14-s − 1.23·16-s − 1.90·17-s − 1.61·19-s + 1.17·23-s + 25-s − 1.90·26-s + 0.236·28-s + 1.90·29-s + 0.726·32-s + 2.23·34-s + 1.90·38-s + 1.17·41-s − 1.38·46-s − 0.618·49-s − 1.17·50-s + 0.618·52-s + 0.449·56-s − 2.23·58-s + 1.90·59-s − 0.618·61-s + 0.381·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5859250980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5859250980\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 1.17T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 + 1.90T + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - 1.17T + T^{2} \) |
| 29 | \( 1 - 1.90T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.17T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.90T + T^{2} \) |
| 61 | \( 1 + 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 + 1.90T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−10.38573377386826066160752646704, −9.112022940416686769446956019113, −8.607630399686697940125649406437, −8.248758297665622738659296317030, −6.90755395880477267551042055507, −6.38157293621589072006005990935, −4.82729322277967008162410716621, −4.15776465182907152916558322417, −2.44799986758828544214896780038, −1.17512664990643477365639614746,
1.17512664990643477365639614746, 2.44799986758828544214896780038, 4.15776465182907152916558322417, 4.82729322277967008162410716621, 6.38157293621589072006005990935, 6.90755395880477267551042055507, 8.248758297665622738659296317030, 8.607630399686697940125649406437, 9.112022940416686769446956019113, 10.38573377386826066160752646704