| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 2·7-s + 2.23·8-s + 3·11-s + 13-s − 1.23·14-s + 1.85·16-s + 0.236·17-s + 6.70·19-s − 1.85·22-s + 7.61·23-s − 0.618·26-s − 3.23·28-s + 1.38·29-s − 4.70·31-s − 5.61·32-s − 0.145·34-s + 2·37-s − 4.14·38-s + 11.6·41-s + 9.61·43-s − 4.85·44-s − 4.70·46-s − 9.23·47-s − 3·49-s − 1.61·52-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.755·7-s + 0.790·8-s + 0.904·11-s + 0.277·13-s − 0.330·14-s + 0.463·16-s + 0.0572·17-s + 1.53·19-s − 0.395·22-s + 1.58·23-s − 0.121·26-s − 0.611·28-s + 0.256·29-s − 0.845·31-s − 0.993·32-s − 0.0250·34-s + 0.328·37-s − 0.672·38-s + 1.81·41-s + 1.46·43-s − 0.731·44-s − 0.694·46-s − 1.34·47-s − 0.428·49-s − 0.224·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781508557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781508557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 + 9.18T + 67T^{2} \) |
| 71 | \( 1 - 1.09T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−8.193882336294560867940576946385, −7.51193252831418462739839390973, −6.98607920434936716189001580065, −5.83080796511311859962397435883, −5.22732470707739054699616299345, −4.47877354462660298716623665938, −3.80172715597284945757639283753, −2.84856787953972731169411469168, −1.44609774381445406962922434299, −0.892707202049531009233107064178,
0.892707202049531009233107064178, 1.44609774381445406962922434299, 2.84856787953972731169411469168, 3.80172715597284945757639283753, 4.47877354462660298716623665938, 5.22732470707739054699616299345, 5.83080796511311859962397435883, 6.98607920434936716189001580065, 7.51193252831418462739839390973, 8.193882336294560867940576946385
