| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 1.61·13-s + 16-s − 0.618·17-s − 18-s − 1.61·26-s + 0.618·29-s − 32-s + 0.618·34-s + 36-s − 0.618·37-s − 1.61·41-s + 49-s + 1.61·52-s + 1.61·53-s − 0.618·58-s − 1.61·61-s + 64-s − 0.618·68-s − 72-s + 1.61·73-s + 0.618·74-s + 81-s + 1.61·82-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 1.61·13-s + 16-s − 0.618·17-s − 18-s − 1.61·26-s + 0.618·29-s − 32-s + 0.618·34-s + 36-s − 0.618·37-s − 1.61·41-s + 49-s + 1.61·52-s + 1.61·53-s − 0.618·58-s − 1.61·61-s + 64-s − 0.618·68-s − 72-s + 1.61·73-s + 0.618·74-s + 81-s + 1.61·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9023899696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9023899696\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 + 0.618T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−8.944191339179056758140172353089, −8.522717546213237955688312177516, −7.67860420595458069790843664791, −6.82384297810429551325557449673, −6.37250204457914593014868612169, −5.36046846446162242401468695755, −4.15073267321779786561828287260, −3.31625537045838323918266477847, −2.04785826340634829589181124630, −1.12162084521037929633987422510,
1.12162084521037929633987422510, 2.04785826340634829589181124630, 3.31625537045838323918266477847, 4.15073267321779786561828287260, 5.36046846446162242401468695755, 6.37250204457914593014868612169, 6.82384297810429551325557449673, 7.67860420595458069790843664791, 8.522717546213237955688312177516, 8.944191339179056758140172353089
