| L(s) = 1 | − 2.36·2-s + 3.58·4-s − 3.75·8-s − 3·9-s + 1.70·16-s + 7.09·18-s − 9.58·23-s − 5·25-s + 10.6·29-s + 3.48·32-s − 10.7·36-s − 1.36·37-s − 8.74·43-s + 22.6·46-s + 11.8·50-s − 13.1·53-s − 25.2·58-s − 11.6·64-s + 16.3·67-s + 9.97·71-s + 11.2·72-s + 3.23·74-s − 12.6·79-s + 9·81-s + 20.6·86-s − 34.3·92-s − 17.9·100-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.79·4-s − 1.32·8-s − 9-s + 0.425·16-s + 1.67·18-s − 1.99·23-s − 25-s + 1.98·29-s + 0.616·32-s − 1.79·36-s − 0.224·37-s − 1.33·43-s + 3.34·46-s + 1.67·50-s − 1.80·53-s − 3.31·58-s − 1.45·64-s + 1.99·67-s + 1.18·71-s + 1.32·72-s + 0.375·74-s − 1.42·79-s + 81-s + 2.22·86-s − 3.58·92-s − 1.79·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4370610869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4370610869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9.58T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 - 9.97T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.191657500028255235866447212254, −7.83025789135503909876025013256, −6.73147352459330420466814695361, −6.30899743965709266700389374617, −5.47450337626528486761660851962, −4.43003800921052062103088679659, −3.34410703578526544534365014623, −2.42876492414158922648548049139, −1.68191468502662224913877117436, −0.43943415230242629590891216453,
0.43943415230242629590891216453, 1.68191468502662224913877117436, 2.42876492414158922648548049139, 3.34410703578526544534365014623, 4.43003800921052062103088679659, 5.47450337626528486761660851962, 6.30899743965709266700389374617, 6.73147352459330420466814695361, 7.83025789135503909876025013256, 8.191657500028255235866447212254
