| L(s) = 1 | + 3-s + 2.93·5-s + 1.35·7-s + 9-s − 0.0643·11-s − 2.07·13-s + 2.93·15-s − 1.73·17-s + 7.35·19-s + 1.35·21-s − 0.641·23-s + 3.60·25-s + 27-s + 3.65·29-s − 3.92·31-s − 0.0643·33-s + 3.97·35-s + 8.28·37-s − 2.07·39-s + 1.55·41-s − 2.77·43-s + 2.93·45-s − 4.51·47-s − 5.15·49-s − 1.73·51-s − 1.55·53-s − 0.188·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.31·5-s + 0.512·7-s + 0.333·9-s − 0.0193·11-s − 0.574·13-s + 0.757·15-s − 0.421·17-s + 1.68·19-s + 0.296·21-s − 0.133·23-s + 0.720·25-s + 0.192·27-s + 0.678·29-s − 0.704·31-s − 0.0111·33-s + 0.672·35-s + 1.36·37-s − 0.331·39-s + 0.242·41-s − 0.423·43-s + 0.437·45-s − 0.658·47-s − 0.737·49-s − 0.243·51-s − 0.212·53-s − 0.0254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.439160182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.439160182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 0.0643T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 23 | \( 1 + 0.641T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 3.92T + 31T^{2} \) |
| 37 | \( 1 - 8.28T + 37T^{2} \) |
| 41 | \( 1 - 1.55T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.16T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.463202031289279606313746472327, −7.76871283307134278589756627926, −7.02224178749497908290951995564, −6.24549598306575649522017577461, −5.35080407644551659479628507937, −4.88547522941156559694750400927, −3.75913834829276337498812270594, −2.73306865778589576369929732117, −2.07753994388778658422917351544, −1.11343893244387602947438037759,
1.11343893244387602947438037759, 2.07753994388778658422917351544, 2.73306865778589576369929732117, 3.75913834829276337498812270594, 4.88547522941156559694750400927, 5.35080407644551659479628507937, 6.24549598306575649522017577461, 7.02224178749497908290951995564, 7.76871283307134278589756627926, 8.463202031289279606313746472327
