| L(s) = 1 | + 2-s − 3-s + 4-s − 2.78·5-s − 6-s − 1.12·7-s + 8-s + 9-s − 2.78·10-s − 11-s − 12-s − 3.29·13-s − 1.12·14-s + 2.78·15-s + 16-s − 3.35·17-s + 18-s + 1.69·19-s − 2.78·20-s + 1.12·21-s − 22-s + 1.69·23-s − 24-s + 2.73·25-s − 3.29·26-s − 27-s − 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.24·5-s − 0.408·6-s − 0.423·7-s + 0.353·8-s + 0.333·9-s − 0.879·10-s − 0.301·11-s − 0.288·12-s − 0.913·13-s − 0.299·14-s + 0.718·15-s + 0.250·16-s − 0.814·17-s + 0.235·18-s + 0.389·19-s − 0.621·20-s + 0.244·21-s − 0.213·22-s + 0.353·23-s − 0.204·24-s + 0.546·25-s − 0.646·26-s − 0.192·27-s − 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221744720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221744720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 + 8.72T + 37T^{2} \) |
| 41 | \( 1 - 3.96T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 0.309T + 71T^{2} \) |
| 73 | \( 1 - 1.36T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−8.221382149954727384393810549958, −7.51443270951948999667888498507, −6.92817850690538933873526531832, −6.29300367796570553862190062614, −5.22399666506496911096859171045, −4.74598281973531628060595804224, −3.92129175322171122278878498843, −3.18923279223383555890260631201, −2.17222365044292610155553002709, −0.55971404403926134796275588977,
0.55971404403926134796275588977, 2.17222365044292610155553002709, 3.18923279223383555890260631201, 3.92129175322171122278878498843, 4.74598281973531628060595804224, 5.22399666506496911096859171045, 6.29300367796570553862190062614, 6.92817850690538933873526531832, 7.51443270951948999667888498507, 8.221382149954727384393810549958
