| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 3·11-s − 12-s + 2·13-s − 15-s + 16-s + 17-s + 18-s + 6·19-s + 20-s − 3·22-s + 2·23-s − 24-s − 4·25-s + 2·26-s − 27-s + 5·29-s − 30-s − 3·31-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 − 0.904·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.639·22-s + 0.417·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.192·27-s + 0.928·29-s − 0.182·30-s − 0.538·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.863051518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863051518\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.033724850588436447587748368036, −7.37554915784802310871377597213, −6.69538936585191546388761044827, −5.76538234977113559945793953404, −5.47458521870123011295881782990, −4.73331249052706002506027037036, −3.76037821253619447184094776283, −2.96966180419138973212091207796, −1.98818309657271369799510489955, −0.874580158156061260992819170398,
0.874580158156061260992819170398, 1.98818309657271369799510489955, 2.96966180419138973212091207796, 3.76037821253619447184094776283, 4.73331249052706002506027037036, 5.47458521870123011295881782990, 5.76538234977113559945793953404, 6.69538936585191546388761044827, 7.37554915784802310871377597213, 8.033724850588436447587748368036
