| L(s) = 1 | + 1.76·2-s + 2.30·3-s + 1.11·4-s + 4.07·6-s − 3.96·7-s − 1.55·8-s + 2.33·9-s − 1.16·11-s + 2.58·12-s − 0.159·13-s − 6.99·14-s − 4.98·16-s + 4.11·18-s + 5.95·19-s − 9.15·21-s − 2.05·22-s + 5.43·23-s − 3.59·24-s − 0.281·26-s − 1.54·27-s − 4.43·28-s + 8.43·29-s + 10.9·31-s − 5.68·32-s − 2.68·33-s + 2.60·36-s + 11.8·37-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 1.33·3-s + 0.559·4-s + 1.66·6-s − 1.49·7-s − 0.550·8-s + 0.776·9-s − 0.350·11-s + 0.745·12-s − 0.0441·13-s − 1.87·14-s − 1.24·16-s + 0.969·18-s + 1.36·19-s − 1.99·21-s − 0.438·22-s + 1.13·23-s − 0.733·24-s − 0.0551·26-s − 0.297·27-s − 0.837·28-s + 1.56·29-s + 1.96·31-s − 1.00·32-s − 0.467·33-s + 0.434·36-s + 1.94·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.011618295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.011618295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 0.159T + 13T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 - 8.43T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 3.99T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 + 7.51T + 53T^{2} \) |
| 59 | \( 1 + 0.790T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 + 2.22T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−7.87565395467453579668337408877, −7.12123460428462573794289693422, −6.38671035545870356014652511915, −5.85375472802643997177780183460, −4.87092722034197567918428384782, −4.24751783818421888833005636280, −3.34385146422980395213941781828, −2.85465669583569355274560269735, −2.64706915150400923887670982512, −0.865050839441749476348211147629,
0.865050839441749476348211147629, 2.64706915150400923887670982512, 2.85465669583569355274560269735, 3.34385146422980395213941781828, 4.24751783818421888833005636280, 4.87092722034197567918428384782, 5.85375472802643997177780183460, 6.38671035545870356014652511915, 7.12123460428462573794289693422, 7.87565395467453579668337408877
