| L(s) = 1 | − 0.311·2-s − 1.90·4-s − 5-s − 4.42·7-s + 1.21·8-s + 0.311·10-s + 2.62·11-s + 0.474·13-s + 1.37·14-s + 3.42·16-s − 5.05·17-s − 19-s + 1.90·20-s − 0.815·22-s + 1.37·23-s + 25-s − 0.147·26-s + 8.42·28-s + 7.80·29-s + 1.24·31-s − 3.49·32-s + 1.57·34-s + 4.42·35-s + 4.47·37-s + 0.311·38-s − 1.21·40-s + 5.05·41-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.951·4-s − 0.447·5-s − 1.67·7-s + 0.429·8-s + 0.0983·10-s + 0.790·11-s + 0.131·13-s + 0.368·14-s + 0.857·16-s − 1.22·17-s − 0.229·19-s + 0.425·20-s − 0.173·22-s + 0.287·23-s + 0.200·25-s − 0.0289·26-s + 1.59·28-s + 1.44·29-s + 0.223·31-s − 0.617·32-s + 0.269·34-s + 0.748·35-s + 0.735·37-s + 0.0504·38-s − 0.192·40-s + 0.788·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7134010042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7134010042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 - 0.474T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 + 7.52T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−10.03582582028932966596141572528, −9.125196098983774754650389017158, −8.885636190164400362017337584355, −7.67690418942320033626894640347, −6.66124902344977942817756341485, −6.00935922428841887255004310514, −4.54675987106302849956955571572, −3.90772688032708116190865295219, −2.80967890591919337285236118589, −0.69766084354299501149010578700,
0.69766084354299501149010578700, 2.80967890591919337285236118589, 3.90772688032708116190865295219, 4.54675987106302849956955571572, 6.00935922428841887255004310514, 6.66124902344977942817756341485, 7.67690418942320033626894640347, 8.885636190164400362017337584355, 9.125196098983774754650389017158, 10.03582582028932966596141572528
