| L(s) = 1 | + 3-s + 4.20·5-s − 3.55·7-s + 9-s + 5.08·11-s + 4.20·15-s + 3.25·17-s − 3.33·19-s − 3.55·21-s + 0.384·23-s + 12.7·25-s + 27-s − 7.28·29-s + 6.47·31-s + 5.08·33-s − 14.9·35-s + 3.12·37-s + 6.77·41-s + 4.15·43-s + 4.20·45-s − 5.49·47-s + 5.64·49-s + 3.25·51-s + 0.613·53-s + 21.3·55-s − 3.33·57-s − 7.87·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·5-s − 1.34·7-s + 0.333·9-s + 1.53·11-s + 1.08·15-s + 0.789·17-s − 0.765·19-s − 0.775·21-s + 0.0800·23-s + 2.54·25-s + 0.192·27-s − 1.35·29-s + 1.16·31-s + 0.884·33-s − 2.52·35-s + 0.514·37-s + 1.05·41-s + 0.634·43-s + 0.627·45-s − 0.801·47-s + 0.805·49-s + 0.456·51-s + 0.0843·53-s + 2.88·55-s − 0.441·57-s − 1.02·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.420423312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.420423312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 - 0.384T + 23T^{2} \) |
| 29 | \( 1 + 7.28T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 0.613T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 2.06T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 3.21T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−8.822821548608283729746967846847, −7.64730757812442081561015318501, −6.69270072190763199013732390754, −6.23974870739884730588494400463, −5.81497510730923109791321172915, −4.63281062565576419054393556719, −3.64874827337870474550179750480, −2.88341094345473856648402181789, −2.02204770549059406244999475312, −1.10740739707672046963647631576,
1.10740739707672046963647631576, 2.02204770549059406244999475312, 2.88341094345473856648402181789, 3.64874827337870474550179750480, 4.63281062565576419054393556719, 5.81497510730923109791321172915, 6.23974870739884730588494400463, 6.69270072190763199013732390754, 7.64730757812442081561015318501, 8.822821548608283729746967846847
