| L(s) = 1 | + 2-s + 0.445·3-s + 4-s + 0.445·6-s + 0.890·7-s + 8-s − 2.80·9-s + 2.75·11-s + 0.445·12-s + 0.890·14-s + 16-s + 2.13·17-s − 2.80·18-s − 0.692·19-s + 0.396·21-s + 2.75·22-s + 0.396·23-s + 0.445·24-s − 2.58·27-s + 0.890·28-s + 0.493·29-s + 7.87·31-s + 32-s + 1.22·33-s + 2.13·34-s − 2.80·36-s − 10.8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.256·3-s + 0.5·4-s + 0.181·6-s + 0.336·7-s + 0.353·8-s − 0.933·9-s + 0.830·11-s + 0.128·12-s + 0.237·14-s + 0.250·16-s + 0.518·17-s − 0.660·18-s − 0.158·19-s + 0.0864·21-s + 0.586·22-s + 0.0825·23-s + 0.0908·24-s − 0.496·27-s + 0.168·28-s + 0.0917·29-s + 1.41·31-s + 0.176·32-s + 0.213·33-s + 0.366·34-s − 0.466·36-s − 1.77·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.892224484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.892224484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 7 | \( 1 - 0.890T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 0.692T + 19T^{2} \) |
| 23 | \( 1 - 0.396T + 23T^{2} \) |
| 29 | \( 1 - 0.493T + 29T^{2} \) |
| 31 | \( 1 - 7.87T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 0.176T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 - 9.30T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 4.67T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−7.935427484476603571980771078652, −6.88467378820070138374473679871, −6.42039638785700222030552196831, −5.61039802244386221162571606298, −5.06328503934263909279014135571, −4.18627031944119399105568439635, −3.53325097971119865777849386507, −2.77427621019138171373937844518, −1.96548068658320401192607824122, −0.869126779356442553161790872867,
0.869126779356442553161790872867, 1.96548068658320401192607824122, 2.77427621019138171373937844518, 3.53325097971119865777849386507, 4.18627031944119399105568439635, 5.06328503934263909279014135571, 5.61039802244386221162571606298, 6.42039638785700222030552196831, 6.88467378820070138374473679871, 7.935427484476603571980771078652
