| L(s) = 1 | + 2-s + 0.414·3-s + 4-s + 5-s + 0.414·6-s + 0.585·7-s + 8-s − 2.82·9-s + 10-s + 2·11-s + 0.414·12-s + 5.65·13-s + 0.585·14-s + 0.414·15-s + 16-s + 6.07·17-s − 2.82·18-s − 3·19-s + 20-s + 0.242·21-s + 2·22-s + 0.414·24-s + 25-s + 5.65·26-s − 2.41·27-s + 0.585·28-s − 5.07·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s + 0.221·7-s + 0.353·8-s − 0.942·9-s + 0.316·10-s + 0.603·11-s + 0.119·12-s + 1.56·13-s + 0.156·14-s + 0.106·15-s + 0.250·16-s + 1.47·17-s − 0.666·18-s − 0.688·19-s + 0.223·20-s + 0.0529·21-s + 0.426·22-s + 0.0845·24-s + 0.200·25-s + 1.10·26-s − 0.464·27-s + 0.110·28-s − 0.941·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.279373025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.279373025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 + 7.48T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 - 2.07T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.118405136000385591531591137512, −7.56411193155271772586968703833, −6.29433168583176606750276530379, −6.14640332262436909893387406197, −5.40182339648774373638705238830, −4.47264338901717540496896140140, −3.59037044088172993961796358026, −3.06956092966487623102972122029, −1.98128238251250156519790483403, −1.06557294274954510422702968714,
1.06557294274954510422702968714, 1.98128238251250156519790483403, 3.06956092966487623102972122029, 3.59037044088172993961796358026, 4.47264338901717540496896140140, 5.40182339648774373638705238830, 6.14640332262436909893387406197, 6.29433168583176606750276530379, 7.56411193155271772586968703833, 8.118405136000385591531591137512
