| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s − 13-s + 16-s − 2·17-s + 18-s − 19-s − 4·22-s − 2·23-s − 24-s − 26-s − 27-s + 4·29-s + 32-s + 4·33-s − 2·34-s + 36-s + 3·37-s − 38-s + 39-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.852·22-s − 0.417·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.742·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 0.160·39-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041979134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041979134\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.893909511025408414009744268015, −6.93880657598779989753774222994, −6.53238272261620612800279204509, −5.59547500730985464441344334863, −5.17656950414959687428838247745, −4.45674095074955608210302249626, −3.69572210888079422895223509190, −2.67227497408484362684533794835, −2.03785763818337133645936034673, −0.64146197935357057367811881592,
0.64146197935357057367811881592, 2.03785763818337133645936034673, 2.67227497408484362684533794835, 3.69572210888079422895223509190, 4.45674095074955608210302249626, 5.17656950414959687428838247745, 5.59547500730985464441344334863, 6.53238272261620612800279204509, 6.93880657598779989753774222994, 7.893909511025408414009744268015
