| L(s) = 1 | + 0.494·2-s − 1.75·4-s − 5-s − 4.28·7-s − 1.85·8-s − 0.494·10-s + 4.48·11-s − 13-s − 2.11·14-s + 2.59·16-s + 1.57·17-s + 1.86·19-s + 1.75·20-s + 2.21·22-s − 5.25·23-s + 25-s − 0.494·26-s + 7.52·28-s − 0.750·29-s + 10.2·31-s + 4.99·32-s + 0.776·34-s + 4.28·35-s + 2.78·37-s + 0.923·38-s + 1.85·40-s − 6.36·41-s + ⋯ |
| L(s) = 1 | + 0.349·2-s − 0.877·4-s − 0.447·5-s − 1.61·7-s − 0.656·8-s − 0.156·10-s + 1.35·11-s − 0.277·13-s − 0.565·14-s + 0.648·16-s + 0.381·17-s + 0.428·19-s + 0.392·20-s + 0.472·22-s − 1.09·23-s + 0.200·25-s − 0.0969·26-s + 1.42·28-s − 0.139·29-s + 1.83·31-s + 0.882·32-s + 0.133·34-s + 0.724·35-s + 0.457·37-s + 0.149·38-s + 0.293·40-s − 0.993·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.494T + 2T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 + 0.750T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 8.27T + 47T^{2} \) |
| 53 | \( 1 - 0.0752T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 - 0.612T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
| 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
−7.927333057436574002903341722540, −6.93929456887041702560235816382, −6.32954449044920999538175701184, −5.78950818648673611399050579475, −4.73559678291759599196200044490, −3.99732296958826979292570497755, −3.49801205621474415745886985966, −2.73608485140440829286033879377, −1.08731239666986688123996083187, 0,
1.08731239666986688123996083187, 2.73608485140440829286033879377, 3.49801205621474415745886985966, 3.99732296958826979292570497755, 4.73559678291759599196200044490, 5.78950818648673611399050579475, 6.32954449044920999538175701184, 6.93929456887041702560235816382, 7.927333057436574002903341722540
