| L(s) = 1 | + 3-s + 5-s + 2.74·7-s + 9-s − 5.76·11-s + 13-s + 15-s − 0.742·17-s + 2.26·19-s + 2.74·21-s + 2.74·23-s + 25-s + 27-s − 0.263·29-s − 5.76·33-s + 2.74·35-s + 7.76·37-s + 39-s − 3.76·41-s − 5.28·43-s + 45-s + 0.521·49-s − 0.742·51-s + 13.0·53-s − 5.76·55-s + 2.26·57-s + 10.0·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.03·7-s + 0.333·9-s − 1.73·11-s + 0.277·13-s + 0.258·15-s − 0.180·17-s + 0.519·19-s + 0.598·21-s + 0.571·23-s + 0.200·25-s + 0.192·27-s − 0.0490·29-s − 1.00·33-s + 0.463·35-s + 1.27·37-s + 0.160·39-s − 0.588·41-s − 0.806·43-s + 0.149·45-s + 0.0744·49-s − 0.103·51-s + 1.79·53-s − 0.777·55-s + 0.299·57-s + 1.30·59-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
| \(L(1)\) |
\(\approx\) |
\(2.999836834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.999836834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 17 | \( 1 + 0.742T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 0.263T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 9.02T + 73T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 0.993T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−8.254933358833290128908754720499, −7.44081859109034377711509422444, −6.83120072221359479545967399189, −5.68432078848289423721347690701, −5.20337203521141644535697029702, −4.55220150750063290231397458897, −3.51969573260954015448618632124, −2.60113212216177839746706904842, −2.04156239707215546311898311221, −0.892706386888282900239307649799,
0.892706386888282900239307649799, 2.04156239707215546311898311221, 2.60113212216177839746706904842, 3.51969573260954015448618632124, 4.55220150750063290231397458897, 5.20337203521141644535697029702, 5.68432078848289423721347690701, 6.83120072221359479545967399189, 7.44081859109034377711509422444, 8.254933358833290128908754720499
