| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s − 2·10-s − 2·12-s + 6·13-s + 2·14-s − 15-s − 4·16-s + 7·17-s − 2·18-s + 8·19-s + 2·20-s + 21-s + 6·23-s − 4·25-s − 12·26-s − 27-s − 2·28-s − 4·29-s + 2·30-s + 2·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 1.66·13-s + 0.534·14-s − 0.258·15-s − 16-s + 1.69·17-s − 0.471·18-s + 1.83·19-s + 0.447·20-s + 0.218·21-s + 1.25·23-s − 4/5·25-s − 2.35·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s + 0.365·30-s + 0.359·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9282747494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9282747494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.137701005835539181857820331023, −8.202583206964865855981890253719, −7.46397791750924418277898456948, −6.87020314373467608834985374187, −5.76084887653943845196075153011, −5.43224410048088467647017343756, −3.95448305592144523815323478185, −3.01461456269301576898447167535, −1.46053632069723485622509197425, −0.898035478225723463309488246787,
0.898035478225723463309488246787, 1.46053632069723485622509197425, 3.01461456269301576898447167535, 3.95448305592144523815323478185, 5.43224410048088467647017343756, 5.76084887653943845196075153011, 6.87020314373467608834985374187, 7.46397791750924418277898456948, 8.202583206964865855981890253719, 9.137701005835539181857820331023
