| L(s) = 1 | − 0.874·2-s − 1.23·4-s + 7-s + 2.82·8-s − 2.28·11-s − 0.236·13-s − 0.874·14-s − 5.99·17-s + 4.47·19-s + 2·22-s + 2.28·23-s + 0.206·26-s − 1.23·28-s + 4.78·29-s − 6.70·31-s − 5.65·32-s + 5.23·34-s + 3·37-s − 3.90·38-s + 2.49·41-s − 3.47·43-s + 2.82·44-s − 2·46-s + 7.94·47-s + 49-s + 0.291·52-s − 3.36·53-s + ⋯ |
| L(s) = 1 | − 0.618·2-s − 0.618·4-s + 0.377·7-s + 0.999·8-s − 0.689·11-s − 0.0654·13-s − 0.233·14-s − 1.45·17-s + 1.02·19-s + 0.426·22-s + 0.477·23-s + 0.0404·26-s − 0.233·28-s + 0.888·29-s − 1.20·31-s − 0.999·32-s + 0.897·34-s + 0.493·37-s − 0.634·38-s + 0.389·41-s − 0.529·43-s + 0.426·44-s − 0.294·46-s + 1.15·47-s + 0.142·49-s + 0.0404·52-s − 0.462·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9350185589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9350185589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 0.874T + 2T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 + 3.36T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 - 7.47T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 1.20T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.331293928913904006869528422505, −7.68462209341477585750458603496, −7.13875439302468502648527086334, −6.13741009240818780306688626740, −5.15149194203808562328924704182, −4.72925136119546174190185947559, −3.84892331276399232117355918798, −2.76910498418566155326963580279, −1.73950244271658019694493077686, −0.59907908323147587804351736040,
0.59907908323147587804351736040, 1.73950244271658019694493077686, 2.76910498418566155326963580279, 3.84892331276399232117355918798, 4.72925136119546174190185947559, 5.15149194203808562328924704182, 6.13741009240818780306688626740, 7.13875439302468502648527086334, 7.68462209341477585750458603496, 8.331293928913904006869528422505
