| L(s) = 1 | − 1.57·2-s + 0.485·4-s + 1.88·5-s − 0.868·7-s + 2.38·8-s − 2.96·10-s + 4.15·11-s − 1.68·13-s + 1.36·14-s − 4.73·16-s − 6.19·17-s − 0.936·19-s + 0.914·20-s − 6.54·22-s − 23-s − 1.45·25-s + 2.65·26-s − 0.421·28-s − 29-s + 5.44·31-s + 2.69·32-s + 9.77·34-s − 1.63·35-s − 9.23·37-s + 1.47·38-s + 4.49·40-s − 8.57·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.242·4-s + 0.841·5-s − 0.328·7-s + 0.844·8-s − 0.938·10-s + 1.25·11-s − 0.467·13-s + 0.366·14-s − 1.18·16-s − 1.50·17-s − 0.214·19-s + 0.204·20-s − 1.39·22-s − 0.208·23-s − 0.291·25-s + 0.521·26-s − 0.0797·28-s − 0.185·29-s + 0.978·31-s + 0.475·32-s + 1.67·34-s − 0.276·35-s − 1.51·37-s + 0.239·38-s + 0.710·40-s − 1.33·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9802392927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9802392927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 0.868T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + 0.936T + 19T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 0.878T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.409011876432150567855969025931, −7.34725687266120192272275813755, −6.76179267915216355250560063889, −6.25224406728859579211130294871, −5.24430596657564532041152000490, −4.42458633535920871720614773724, −3.68898524603152392939712864710, −2.30915054002250223885310225836, −1.78048729446068969283411079830, −0.62058253710722327872943916501,
0.62058253710722327872943916501, 1.78048729446068969283411079830, 2.30915054002250223885310225836, 3.68898524603152392939712864710, 4.42458633535920871720614773724, 5.24430596657564532041152000490, 6.25224406728859579211130294871, 6.76179267915216355250560063889, 7.34725687266120192272275813755, 8.409011876432150567855969025931
