| L(s) = 1 | + 2.30·3-s + 4.32·5-s − 4.01·7-s + 2.30·9-s + 3.81·11-s − 13-s + 9.97·15-s + 1.35·17-s + 7.68·19-s − 9.25·21-s − 6.30·23-s + 13.7·25-s − 1.60·27-s + 29-s − 3.86·31-s + 8.79·33-s − 17.3·35-s + 11.5·37-s − 2.30·39-s − 5.85·41-s + 10.6·43-s + 9.97·45-s − 8.14·47-s + 9.13·49-s + 3.11·51-s + 1.83·53-s + 16.5·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 1.93·5-s − 1.51·7-s + 0.768·9-s + 1.15·11-s − 0.277·13-s + 2.57·15-s + 0.328·17-s + 1.76·19-s − 2.01·21-s − 1.31·23-s + 2.74·25-s − 0.308·27-s + 0.185·29-s − 0.693·31-s + 1.53·33-s − 2.93·35-s + 1.89·37-s − 0.368·39-s − 0.914·41-s + 1.62·43-s + 1.48·45-s − 1.18·47-s + 1.30·49-s + 0.436·51-s + 0.252·53-s + 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.549649697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.549649697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 4.32T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 1.83T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 - 0.164T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 4.43T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−8.230183847212430009522733218976, −7.26440106101944741416415232134, −6.67338574642846390701595039162, −5.91906748306663895896915855040, −5.53468007567745885173716388208, −4.19158484860356768004899521161, −3.33233395785235177143517991926, −2.80149744128283434166297755302, −2.05588633466572585206506097438, −1.11921011584976000299261301798,
1.11921011584976000299261301798, 2.05588633466572585206506097438, 2.80149744128283434166297755302, 3.33233395785235177143517991926, 4.19158484860356768004899521161, 5.53468007567745885173716388208, 5.91906748306663895896915855040, 6.67338574642846390701595039162, 7.26440106101944741416415232134, 8.230183847212430009522733218976
