| L(s) = 1 | − 2.15·2-s − 0.591·3-s + 2.65·4-s + 1.27·6-s − 2.52·7-s − 1.42·8-s − 2.64·9-s − 4.22·11-s − 1.57·12-s − 0.0216·13-s + 5.44·14-s − 2.24·16-s + 6.10·17-s + 5.71·18-s − 6.77·19-s + 1.49·21-s + 9.12·22-s − 0.634·23-s + 0.841·24-s + 0.0467·26-s + 3.34·27-s − 6.70·28-s − 5.11·29-s + 9.22·31-s + 7.69·32-s + 2.50·33-s − 13.1·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.341·3-s + 1.32·4-s + 0.521·6-s − 0.953·7-s − 0.502·8-s − 0.883·9-s − 1.27·11-s − 0.454·12-s − 0.00601·13-s + 1.45·14-s − 0.561·16-s + 1.48·17-s + 1.34·18-s − 1.55·19-s + 0.325·21-s + 1.94·22-s − 0.132·23-s + 0.171·24-s + 0.00917·26-s + 0.643·27-s − 1.26·28-s − 0.950·29-s + 1.65·31-s + 1.36·32-s + 0.435·33-s − 2.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.591T + 3T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 13 | \( 1 + 0.0216T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 + 0.634T + 23T^{2} \) |
| 29 | \( 1 + 5.11T + 29T^{2} \) |
| 31 | \( 1 - 9.22T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 + 0.605T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 + 3.78T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
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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
−8.093444669584743049809001043836, −7.46028297211383985133502232178, −6.52547142548146407184052382762, −5.97988538159650874933747048072, −5.20636064529407338779448574941, −4.06960268555075620503901596809, −2.86481870738525701283961124647, −2.35281884189311050098561322123, −0.865390618173994129190014340837, 0,
0.865390618173994129190014340837, 2.35281884189311050098561322123, 2.86481870738525701283961124647, 4.06960268555075620503901596809, 5.20636064529407338779448574941, 5.97988538159650874933747048072, 6.52547142548146407184052382762, 7.46028297211383985133502232178, 8.093444669584743049809001043836
