| L(s) = 1 | − 0.936·3-s + 1.25·5-s − 0.290·7-s − 2.12·9-s − 1.70·11-s − 2.27·13-s − 1.17·15-s − 4.69·17-s + 0.709·19-s + 0.271·21-s + 3.51·23-s − 3.42·25-s + 4.79·27-s + 4.26·29-s − 0.767·31-s + 1.59·33-s − 0.364·35-s − 6.00·37-s + 2.13·39-s + 4.17·41-s − 8.76·43-s − 2.66·45-s + 6.27·47-s − 6.91·49-s + 4.39·51-s − 7.14·53-s − 2.14·55-s + ⋯ |
| L(s) = 1 | − 0.540·3-s + 0.561·5-s − 0.109·7-s − 0.707·9-s − 0.515·11-s − 0.631·13-s − 0.303·15-s − 1.13·17-s + 0.162·19-s + 0.0593·21-s + 0.733·23-s − 0.684·25-s + 0.922·27-s + 0.791·29-s − 0.137·31-s + 0.278·33-s − 0.0616·35-s − 0.988·37-s + 0.341·39-s + 0.651·41-s − 1.33·43-s − 0.397·45-s + 0.915·47-s − 0.987·49-s + 0.615·51-s − 0.981·53-s − 0.289·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.006276470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006276470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.936T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 0.290T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 - 0.709T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 0.767T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 4.17T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 0.666T + 79T^{2} \) |
| 83 | \( 1 - 5.50T + 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
| 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
−7.85086042785001558388461697686, −6.92961017024413632251175636991, −6.46477175680829794329932806149, −5.70370083619873601147726664165, −5.10188762467641315451759306093, −4.55630274965162388279265148358, −3.38176810004608607957771244499, −2.61505170392244362947787438133, −1.85851536114281520175687351961, −0.48680498627401129318022689546,
0.48680498627401129318022689546, 1.85851536114281520175687351961, 2.61505170392244362947787438133, 3.38176810004608607957771244499, 4.55630274965162388279265148358, 5.10188762467641315451759306093, 5.70370083619873601147726664165, 6.46477175680829794329932806149, 6.92961017024413632251175636991, 7.85086042785001558388461697686
