| L(s) = 1 | − 2-s + 3-s + 4-s + 4.27·5-s − 6-s − 8-s + 9-s − 4.27·10-s − 2.27·11-s + 12-s + 13-s + 4.27·15-s + 16-s − 0.274·17-s − 18-s + 2.27·19-s + 4.27·20-s + 2.27·22-s + 2.27·23-s − 24-s + 13.2·25-s − 26-s + 27-s + 8.27·29-s − 4.27·30-s − 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.91·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.35·10-s − 0.685·11-s + 0.288·12-s + 0.277·13-s + 1.10·15-s + 0.250·16-s − 0.0666·17-s − 0.235·18-s + 0.521·19-s + 0.955·20-s + 0.485·22-s + 0.474·23-s − 0.204·24-s + 2.65·25-s − 0.196·26-s + 0.192·27-s + 1.53·29-s − 0.780·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.590456619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.590456619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 + 0.274T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.632718934712415615076700767581, −7.974300782950850933387185747782, −6.88517958231511771567876718046, −6.54710810803300730587817640424, −5.42521124514534574699953283675, −5.10741852580786120667089600485, −3.53998009425476247684296362885, −2.62025861886281616824784174913, −2.01144168489934844065351739579, −1.06710519062030289478825316279,
1.06710519062030289478825316279, 2.01144168489934844065351739579, 2.62025861886281616824784174913, 3.53998009425476247684296362885, 5.10741852580786120667089600485, 5.42521124514534574699953283675, 6.54710810803300730587817640424, 6.88517958231511771567876718046, 7.974300782950850933387185747782, 8.632718934712415615076700767581
