| L(s) = 1 | + 2.30·3-s − 0.334·5-s − 0.428·7-s + 2.31·9-s − 13-s − 0.771·15-s + 5.96·17-s − 5.76·19-s − 0.986·21-s − 8.85·23-s − 4.88·25-s − 1.58·27-s + 1.24·29-s − 6.79·31-s + 0.143·35-s − 2.35·37-s − 2.30·39-s + 6.51·41-s + 8.32·43-s − 0.773·45-s + 1.80·47-s − 6.81·49-s + 13.7·51-s − 11.6·53-s − 13.2·57-s − 2.92·59-s − 14.0·61-s + ⋯ |
| L(s) = 1 | + 1.33·3-s − 0.149·5-s − 0.161·7-s + 0.770·9-s − 0.277·13-s − 0.199·15-s + 1.44·17-s − 1.32·19-s − 0.215·21-s − 1.84·23-s − 0.977·25-s − 0.305·27-s + 0.231·29-s − 1.22·31-s + 0.0242·35-s − 0.387·37-s − 0.369·39-s + 1.01·41-s + 1.26·43-s − 0.115·45-s + 0.263·47-s − 0.973·49-s + 1.92·51-s − 1.59·53-s − 1.75·57-s − 0.380·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 0.334T + 5T^{2} \) |
| 7 | \( 1 + 0.428T + 7T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 + 6.79T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 7.21T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 0.324T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
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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
−7.936429323400447125468792043639, −7.31418901752524684152821163749, −6.18117583837881086055378807634, −5.73117215724443185128712497424, −4.52003462617674335957685896000, −3.84417322020591045704400109325, −3.25050669373935549395653342077, −2.32618269914534873096097666238, −1.67052941978362379193701671336, 0,
1.67052941978362379193701671336, 2.32618269914534873096097666238, 3.25050669373935549395653342077, 3.84417322020591045704400109325, 4.52003462617674335957685896000, 5.73117215724443185128712497424, 6.18117583837881086055378807634, 7.31418901752524684152821163749, 7.936429323400447125468792043639
