| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s + 9-s − 3·10-s − 12-s + 13-s − 3·15-s + 16-s − 17-s − 18-s + 4·19-s + 3·20-s + 6·23-s + 24-s + 4·25-s − 26-s − 27-s + 3·29-s + 3·30-s + 31-s − 32-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s + 0.277·13-s − 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.670·20-s + 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.557·29-s + 0.547·30-s + 0.179·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-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
| \(L(1)\) |
\(\approx\) |
\(1.625254511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625254511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.413798308651287722933320258985, −7.41301777746684554464209249671, −6.73647706113444287626283522031, −6.21679148678412444131220994181, −5.37701209793317614244231344748, −4.93023859411002440005930250448, −3.56876179022236201632569459867, −2.60369354388022592332699224772, −1.68632594601745114664182572333, −0.841452018683176692211816231500,
0.841452018683176692211816231500, 1.68632594601745114664182572333, 2.60369354388022592332699224772, 3.56876179022236201632569459867, 4.93023859411002440005930250448, 5.37701209793317614244231344748, 6.21679148678412444131220994181, 6.73647706113444287626283522031, 7.41301777746684554464209249671, 8.413798308651287722933320258985
