| L(s) = 1 | + 3.08·3-s + 3.25·5-s + 6.53·9-s − 5.12·11-s + 5.13·13-s + 10.0·15-s − 17-s − 5.25·19-s + 6.97·23-s + 5.59·25-s + 10.9·27-s − 3.39·29-s + 2.35·31-s − 15.8·33-s + 2.60·37-s + 15.8·39-s − 5.70·41-s + 1.18·43-s + 21.2·45-s + 4.70·47-s − 3.08·51-s + 10.0·53-s − 16.6·55-s − 16.2·57-s − 13.0·59-s + 1.78·61-s + 16.7·65-s + ⋯ |
| L(s) = 1 | + 1.78·3-s + 1.45·5-s + 2.17·9-s − 1.54·11-s + 1.42·13-s + 2.59·15-s − 0.242·17-s − 1.20·19-s + 1.45·23-s + 1.11·25-s + 2.10·27-s − 0.630·29-s + 0.423·31-s − 2.75·33-s + 0.428·37-s + 2.53·39-s − 0.891·41-s + 0.180·43-s + 3.17·45-s + 0.685·47-s − 0.432·51-s + 1.37·53-s − 2.24·55-s − 2.14·57-s − 1.70·59-s + 0.228·61-s + 2.07·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.772073274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.772073274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 + 3.39T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.684150560448497575617921535116, −8.145645810533883411236497306271, −7.27681623740452755856804976266, −6.45007690567173264611580967833, −5.62982434307992907307550230988, −4.71717866697391026894789536632, −3.69452538797608346830589027071, −2.77501668657397723823843904395, −2.27741904580852919683964362800, −1.37580546528844004142723143907,
1.37580546528844004142723143907, 2.27741904580852919683964362800, 2.77501668657397723823843904395, 3.69452538797608346830589027071, 4.71717866697391026894789536632, 5.62982434307992907307550230988, 6.45007690567173264611580967833, 7.27681623740452755856804976266, 8.145645810533883411236497306271, 8.684150560448497575617921535116
