L(s) = 1 | + 2.13·2-s + 2.23·3-s + 2.54·4-s + 4.75·6-s − 1.48·7-s + 1.16·8-s + 1.97·9-s + 4.68·11-s + 5.68·12-s − 0.361·13-s − 3.16·14-s − 2.60·16-s + 5.47·17-s + 4.21·18-s − 3.31·21-s + 9.98·22-s + 6.16·23-s + 2.60·24-s − 0.769·26-s − 2.28·27-s − 3.78·28-s + 0.895·29-s − 4.80·31-s − 7.89·32-s + 10.4·33-s + 11.6·34-s + 5.02·36-s + ⋯ |
L(s) = 1 | + 1.50·2-s + 1.28·3-s + 1.27·4-s + 1.94·6-s − 0.561·7-s + 0.412·8-s + 0.658·9-s + 1.41·11-s + 1.63·12-s − 0.100·13-s − 0.846·14-s − 0.651·16-s + 1.32·17-s + 0.992·18-s − 0.723·21-s + 2.12·22-s + 1.28·23-s + 0.531·24-s − 0.150·26-s − 0.440·27-s − 0.715·28-s + 0.166·29-s − 0.862·31-s − 1.39·32-s + 1.81·33-s + 2.00·34-s + 0.838·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.424515602\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.424515602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 + 0.361T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 0.895T + 29T^{2} \) |
| 31 | \( 1 + 4.80T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.52T + 53T^{2} \) |
| 59 | \( 1 - 9.80T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 - 8.85T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 - 3.32T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−7.51723150280944624346215433470, −6.99506272389604494319247488628, −6.21508109680371900657605140720, −5.66887087839387715422570052084, −4.77304536633881588656759909386, −3.99187835630352805437099911388, −3.48569726050423765640089102074, −2.99555963687108821725985108102, −2.23566997070555733694348710518, −1.10791784860229896630846604771,
1.10791784860229896630846604771, 2.23566997070555733694348710518, 2.99555963687108821725985108102, 3.48569726050423765640089102074, 3.99187835630352805437099911388, 4.77304536633881588656759909386, 5.66887087839387715422570052084, 6.21508109680371900657605140720, 6.99506272389604494319247488628, 7.51723150280944624346215433470