| L(s) = 1 | − 4.93·2-s + 9.92·3-s + 16.3·4-s − 48.9·6-s + 14.6·7-s − 41.3·8-s + 71.4·9-s + 49.6·11-s + 162.·12-s + 59.3·13-s − 72.2·14-s + 73.3·16-s + 4.26·17-s − 352.·18-s − 2.76·19-s + 145.·21-s − 245.·22-s + 23·23-s − 410.·24-s − 293.·26-s + 441.·27-s + 239.·28-s − 180.·29-s − 131.·31-s − 30.9·32-s + 492.·33-s − 21.0·34-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.90·3-s + 2.04·4-s − 3.33·6-s + 0.789·7-s − 1.82·8-s + 2.64·9-s + 1.36·11-s + 3.91·12-s + 1.26·13-s − 1.37·14-s + 1.14·16-s + 0.0609·17-s − 4.62·18-s − 0.0333·19-s + 1.50·21-s − 2.37·22-s + 0.208·23-s − 3.49·24-s − 2.21·26-s + 3.14·27-s + 1.61·28-s − 1.15·29-s − 0.761·31-s − 0.170·32-s + 2.59·33-s − 0.106·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.378514168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378514168\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 4.93T + 8T^{2} \) |
| 3 | \( 1 - 9.92T + 27T^{2} \) |
| 7 | \( 1 - 14.6T + 343T^{2} \) |
| 11 | \( 1 - 49.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.26T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.76T + 6.85e3T^{2} \) |
| 29 | \( 1 + 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 380.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 97.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 799.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 227.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 59.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 132.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−9.858875143672564373346568979209, −9.079264807648853999868609041810, −8.788677196636899316602933072001, −7.958799213979819391396550046866, −7.35878052820082569967920409076, −6.35815455787118478332155056021, −4.21190592877074552258209427865, −3.19128474603648983221505562804, −1.80367635054241836360647873147, −1.32302460024584306239009722230,
1.32302460024584306239009722230, 1.80367635054241836360647873147, 3.19128474603648983221505562804, 4.21190592877074552258209427865, 6.35815455787118478332155056021, 7.35878052820082569967920409076, 7.958799213979819391396550046866, 8.788677196636899316602933072001, 9.079264807648853999868609041810, 9.858875143672564373346568979209
