| L(s) = 1 | − 2.79·3-s − 0.127·7-s + 4.83·9-s + 5.21·11-s + 0.515·13-s − 3.58·17-s + 19-s + 0.357·21-s + 6.50·23-s − 5.14·27-s − 3.05·29-s + 5.44·31-s − 14.5·33-s + 4.99·37-s − 1.44·39-s + 11.4·41-s + 2.06·43-s − 11.9·47-s − 6.98·49-s + 10.0·51-s − 2.25·53-s − 2.79·57-s + 1.89·59-s − 5.83·61-s − 0.616·63-s − 0.432·67-s − 18.2·69-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 0.0482·7-s + 1.61·9-s + 1.57·11-s + 0.142·13-s − 0.868·17-s + 0.229·19-s + 0.0779·21-s + 1.35·23-s − 0.989·27-s − 0.567·29-s + 0.977·31-s − 2.54·33-s + 0.821·37-s − 0.231·39-s + 1.78·41-s + 0.315·43-s − 1.73·47-s − 0.997·49-s + 1.40·51-s − 0.310·53-s − 0.370·57-s + 0.246·59-s − 0.746·61-s − 0.0777·63-s − 0.0528·67-s − 2.19·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130277600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130277600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 + 0.127T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 - 0.515T + 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 2.06T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 + 5.83T + 61T^{2} \) |
| 67 | \( 1 + 0.432T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 9.98T + 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 0.496T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−8.608745651008628304098962327464, −7.49163907542998302063450450876, −6.72102504178230645985307422390, −6.33086137809520624732184850792, −5.64811165387960118728315524922, −4.65869862139211995941945149502, −4.26604169880664857795514823217, −3.06957168552183725983460075786, −1.58445090308576382903331718354, −0.71925271457209539483102620124,
0.71925271457209539483102620124, 1.58445090308576382903331718354, 3.06957168552183725983460075786, 4.26604169880664857795514823217, 4.65869862139211995941945149502, 5.64811165387960118728315524922, 6.33086137809520624732184850792, 6.72102504178230645985307422390, 7.49163907542998302063450450876, 8.608745651008628304098962327464
