L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 3·11-s − 13-s − 15-s − 5·17-s + 2·19-s + 21-s + 3·23-s + 25-s − 27-s + 4·31-s − 3·33-s − 35-s + 37-s + 39-s + 9·41-s − 2·43-s + 45-s − 8·47-s − 6·49-s + 5·51-s − 53-s + 3·55-s − 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 0.458·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 0.164·37-s + 0.160·39-s + 1.40·41-s − 0.304·43-s + 0.149·45-s − 1.16·47-s − 6/7·49-s + 0.700·51-s − 0.137·53-s + 0.404·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.640360467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.640360467\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.032738011730645770768761631572, −7.10478351307535636375613719077, −6.52872554201688808231341911458, −6.12132471512271349064332017064, −5.13133879437452927879317491456, −4.56802817050921818408484136696, −3.68529075760365404601955128362, −2.73148715599548466722492138220, −1.75712103104138674899765724412, −0.69874911503415318511062226331,
0.69874911503415318511062226331, 1.75712103104138674899765724412, 2.73148715599548466722492138220, 3.68529075760365404601955128362, 4.56802817050921818408484136696, 5.13133879437452927879317491456, 6.12132471512271349064332017064, 6.52872554201688808231341911458, 7.10478351307535636375613719077, 8.032738011730645770768761631572