| L(s) = 1 | − 1.56·3-s + 3.56·5-s − 3.12·7-s − 0.561·9-s + 5.12·13-s − 5.56·15-s − 2·17-s + 4·19-s + 4.87·21-s + 2.43·23-s + 7.68·25-s + 5.56·27-s + 5.12·29-s − 5.56·31-s − 11.1·35-s − 7.56·37-s − 8·39-s + 1.12·41-s + 7.12·43-s − 2·45-s + 8·47-s + 2.75·49-s + 3.12·51-s + 12.2·53-s − 6.24·57-s + 7.80·59-s − 1.12·61-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 1.59·5-s − 1.18·7-s − 0.187·9-s + 1.42·13-s − 1.43·15-s − 0.485·17-s + 0.917·19-s + 1.06·21-s + 0.508·23-s + 1.53·25-s + 1.07·27-s + 0.951·29-s − 0.998·31-s − 1.88·35-s − 1.24·37-s − 1.28·39-s + 0.175·41-s + 1.08·43-s − 0.298·45-s + 1.16·47-s + 0.393·49-s + 0.437·51-s + 1.68·53-s − 0.827·57-s + 1.01·59-s − 0.143·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.384585135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384585135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−10.19755635590132717366414803194, −9.164341463006006017680152376450, −8.771660399983008132430193928192, −7.07560819296462536396012593140, −6.34369526057879782881551196050, −5.82213997211826838360050364415, −5.17664716148707366578576598529, −3.61632211414597521079215968222, −2.50431074557952801149547415729, −0.999881244511981930607956401069,
0.999881244511981930607956401069, 2.50431074557952801149547415729, 3.61632211414597521079215968222, 5.17664716148707366578576598529, 5.82213997211826838360050364415, 6.34369526057879782881551196050, 7.07560819296462536396012593140, 8.771660399983008132430193928192, 9.164341463006006017680152376450, 10.19755635590132717366414803194
