L(s) = 1 | + 4·2-s + 30.2·3-s + 16·4-s − 25·5-s + 121.·6-s + 98.1·7-s + 64·8-s + 672.·9-s − 100·10-s − 163.·11-s + 484.·12-s + 1.03e3·13-s + 392.·14-s − 756.·15-s + 256·16-s − 1.42e3·17-s + 2.69e3·18-s − 2.25e3·19-s − 400·20-s + 2.96e3·21-s − 655.·22-s − 529·23-s + 1.93e3·24-s + 625·25-s + 4.15e3·26-s + 1.30e4·27-s + 1.56e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.94·3-s + 0.5·4-s − 0.447·5-s + 1.37·6-s + 0.756·7-s + 0.353·8-s + 2.76·9-s − 0.316·10-s − 0.408·11-s + 0.970·12-s + 1.70·13-s + 0.535·14-s − 0.868·15-s + 0.250·16-s − 1.19·17-s + 1.95·18-s − 1.43·19-s − 0.223·20-s + 1.46·21-s − 0.288·22-s − 0.208·23-s + 0.686·24-s + 0.200·25-s + 1.20·26-s + 3.43·27-s + 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(6.555664855\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.555664855\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 + 529T \) |
good | 3 | \( 1 - 30.2T + 243T^{2} \) |
| 7 | \( 1 - 98.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.25e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 52.3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.07e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 157.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.13e4T + 8.58e9T^{2} \) |
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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
−11.28636054369201021942078455589, −10.48664905674646028183154986570, −8.978822559104636984854062100696, −8.372649303821137306006596346531, −7.59982105153546538816570808059, −6.37323037544805708505200801078, −4.45555710774854166739011793000, −3.86863879226278006217442717516, −2.61658286569381946533488366646, −1.59078820028914841616248654923,
1.59078820028914841616248654923, 2.61658286569381946533488366646, 3.86863879226278006217442717516, 4.45555710774854166739011793000, 6.37323037544805708505200801078, 7.59982105153546538816570808059, 8.372649303821137306006596346531, 8.978822559104636984854062100696, 10.48664905674646028183154986570, 11.28636054369201021942078455589