| L(s) = 1 | + 2-s − 2.45·3-s + 4-s − 3.68·5-s − 2.45·6-s + 8-s + 3.01·9-s − 3.68·10-s − 6.14·11-s − 2.45·12-s − 1.79·13-s + 9.03·15-s + 16-s + 2.08·17-s + 3.01·18-s + 3.98·19-s − 3.68·20-s − 6.14·22-s + 6.96·23-s − 2.45·24-s + 8.59·25-s − 1.79·26-s − 0.0257·27-s − 8.33·29-s + 9.03·30-s − 7.88·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 1.64·5-s − 1.00·6-s + 0.353·8-s + 1.00·9-s − 1.16·10-s − 1.85·11-s − 0.707·12-s − 0.497·13-s + 2.33·15-s + 0.250·16-s + 0.504·17-s + 0.709·18-s + 0.914·19-s − 0.824·20-s − 1.31·22-s + 1.45·23-s − 0.500·24-s + 1.71·25-s − 0.351·26-s − 0.00495·27-s − 1.54·29-s + 1.65·30-s − 1.41·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 - 2.85T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 0.480T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
| 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
−7.28745635270481376309425753785, −7.06984761046187135240646074907, −5.74792307987324244001743919219, −5.31153383710796334422546463060, −4.95575994713310566204293093549, −4.03513752269030206967900701730, −3.33465003939997517142904555664, −2.48409391252027099428427775717, −0.876479679807656091175189459637, 0,
0.876479679807656091175189459637, 2.48409391252027099428427775717, 3.33465003939997517142904555664, 4.03513752269030206967900701730, 4.95575994713310566204293093549, 5.31153383710796334422546463060, 5.74792307987324244001743919219, 7.06984761046187135240646074907, 7.28745635270481376309425753785
