| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 4·11-s − 12-s + 2·13-s + 15-s + 16-s + 2·17-s + 18-s + 19-s − 20-s − 4·22-s − 8·23-s − 24-s + 25-s + 2·26-s − 27-s + 6·29-s + 30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944112451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944112451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.40253562720608, −14.59889718256606, −14.15494669979071, −13.58191749190117, −13.11478901609828, −12.38886946902008, −12.16146585461082, −11.63171603883111, −10.87304998308303, −10.55590271226215, −10.07851469762837, −9.332560234450665, −8.333961099017157, −8.139908308250313, −7.245651332468497, −7.004036509333565, −5.938934884571963, −5.735278555799318, −5.118070682194898, −4.335539369273602, −3.890970950818089, −3.108010125600276, −2.406003593737514, −1.529853888891279, −0.4978234782573722,
0.4978234782573722, 1.529853888891279, 2.406003593737514, 3.108010125600276, 3.890970950818089, 4.335539369273602, 5.118070682194898, 5.735278555799318, 5.938934884571963, 7.004036509333565, 7.245651332468497, 8.139908308250313, 8.333961099017157, 9.332560234450665, 10.07851469762837, 10.55590271226215, 10.87304998308303, 11.63171603883111, 12.16146585461082, 12.38886946902008, 13.11478901609828, 13.58191749190117, 14.15494669979071, 14.59889718256606, 15.40253562720608
