| L(s) = 1 | − 5-s − 2.82·7-s + 4.94·11-s + 4.65·13-s + 2.16·17-s − 0.347·19-s − 23-s + 25-s − 1.32·29-s + 10.0·31-s + 2.82·35-s − 5.95·37-s + 4.98·41-s + 1.57·43-s − 13.0·47-s + 1.00·49-s − 5.12·53-s − 4.94·55-s + 12.4·59-s + 10.4·61-s − 4.65·65-s − 12.2·67-s − 4.84·71-s + 12.3·73-s − 13.9·77-s − 8.81·79-s + 14.0·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.06·7-s + 1.49·11-s + 1.29·13-s + 0.525·17-s − 0.0798·19-s − 0.208·23-s + 0.200·25-s − 0.245·29-s + 1.80·31-s + 0.478·35-s − 0.978·37-s + 0.778·41-s + 0.240·43-s − 1.89·47-s + 0.144·49-s − 0.703·53-s − 0.666·55-s + 1.61·59-s + 1.33·61-s − 0.577·65-s − 1.49·67-s − 0.575·71-s + 1.44·73-s − 1.59·77-s − 0.991·79-s + 1.54·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911069458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911069458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 19 | \( 1 + 0.347T + 19T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 - 4.98T + 41T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.84T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 8.81T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.889596061241644236971422046840, −6.89601636619348690697070695677, −6.44090745853289084907801876701, −6.01777860265393794521555888432, −4.96780896673215715667017249072, −3.98584123027292345588302471878, −3.62697526250809974222098268465, −2.89315226302397280941841447118, −1.58667384438298413357274807031, −0.72027251833946690790423496568,
0.72027251833946690790423496568, 1.58667384438298413357274807031, 2.89315226302397280941841447118, 3.62697526250809974222098268465, 3.98584123027292345588302471878, 4.96780896673215715667017249072, 6.01777860265393794521555888432, 6.44090745853289084907801876701, 6.89601636619348690697070695677, 7.889596061241644236971422046840
