| L(s) = 1 | − 1.87·2-s + 2.53·4-s − 1.87·5-s − 7-s − 2.87·8-s + 9-s + 3.53·10-s + 0.347·11-s + 1.87·14-s + 2.87·16-s − 1.87·18-s − 1.87·19-s − 4.75·20-s − 0.652·22-s − 23-s + 2.53·25-s − 2.53·28-s + 0.347·29-s − 2.53·32-s + 1.87·35-s + 2.53·36-s + 2·37-s + 3.53·38-s + 5.41·40-s + 0.347·41-s + 0.879·44-s − 1.87·45-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.53·4-s − 1.87·5-s − 7-s − 2.87·8-s + 9-s + 3.53·10-s + 0.347·11-s + 1.87·14-s + 2.87·16-s − 1.87·18-s − 1.87·19-s − 4.75·20-s − 0.652·22-s − 23-s + 2.53·25-s − 2.53·28-s + 0.347·29-s − 2.53·32-s + 1.87·35-s + 2.53·36-s + 2·37-s + 3.53·38-s + 5.41·40-s + 0.347·41-s + 0.879·44-s − 1.87·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2551935557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2551935557\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1399 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.87T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2T + T^{2} \) |
| 41 | \( 1 - 0.347T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - 1.53T + T^{2} \) |
| 89 | \( 1 - 1.53T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.644783436768664540910238586277, −8.941717319702858673938916501755, −8.036611859775545535971489951826, −7.74608267319443756917231089509, −6.68557149873679480461627036315, −6.43773220657737047033485917765, −4.34430281478938848373164255729, −3.59764425696221579862302738368, −2.31940306157713862423758059803, −0.68732815282762123200203408539,
0.68732815282762123200203408539, 2.31940306157713862423758059803, 3.59764425696221579862302738368, 4.34430281478938848373164255729, 6.43773220657737047033485917765, 6.68557149873679480461627036315, 7.74608267319443756917231089509, 8.036611859775545535971489951826, 8.941717319702858673938916501755, 9.644783436768664540910238586277
