| L(s) = 1 | − 5-s − 4.83·7-s − 2.25·11-s − 13-s − 6.32·17-s − 2.58·19-s − 4.83·23-s + 25-s − 9.09·29-s − 0.510·31-s + 4.83·35-s + 4.25·37-s − 0.255·41-s + 9.16·43-s + 4.51·47-s + 16.4·49-s + 9.93·53-s + 2.25·55-s − 14.1·59-s + 9.93·61-s + 65-s − 13.1·67-s + 5.74·71-s + 2.90·73-s + 10.9·77-s + 5.74·79-s − 10.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.82·7-s − 0.679·11-s − 0.277·13-s − 1.53·17-s − 0.592·19-s − 1.00·23-s + 0.200·25-s − 1.68·29-s − 0.0916·31-s + 0.818·35-s + 0.699·37-s − 0.0398·41-s + 1.39·43-s + 0.657·47-s + 2.34·49-s + 1.36·53-s + 0.304·55-s − 1.84·59-s + 1.27·61-s + 0.124·65-s − 1.60·67-s + 0.681·71-s + 0.340·73-s + 1.24·77-s + 0.646·79-s − 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3956408323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3956408323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 + 0.510T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 0.255T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.307570557854964803566252735020, −7.45059995352314743278744328710, −6.89977806415916137912556072151, −6.13484908617887122548345975936, −5.57686208155334947777720405417, −4.30702375748260935994724650855, −3.87369940944715429293985538171, −2.83528862120194662445192562441, −2.19200091241778254745470924294, −0.32002839491010516913047561627,
0.32002839491010516913047561627, 2.19200091241778254745470924294, 2.83528862120194662445192562441, 3.87369940944715429293985538171, 4.30702375748260935994724650855, 5.57686208155334947777720405417, 6.13484908617887122548345975936, 6.89977806415916137912556072151, 7.45059995352314743278744328710, 8.307570557854964803566252735020
