| L(s) = 1 | + 1.53·2-s − 3.17·3-s + 0.369·4-s − 5-s − 4.87·6-s − 7-s − 2.51·8-s + 7.04·9-s − 1.53·10-s − 1.17·12-s + 0.829·13-s − 1.53·14-s + 3.17·15-s − 4.60·16-s + 2.82·17-s + 10.8·18-s − 6.49·19-s − 0.369·20-s + 3.17·21-s − 6.97·23-s + 7.95·24-s + 25-s + 1.27·26-s − 12.8·27-s − 0.369·28-s − 3.26·29-s + 4.87·30-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 1.83·3-s + 0.184·4-s − 0.447·5-s − 1.99·6-s − 0.377·7-s − 0.887·8-s + 2.34·9-s − 0.486·10-s − 0.337·12-s + 0.230·13-s − 0.411·14-s + 0.818·15-s − 1.15·16-s + 0.686·17-s + 2.55·18-s − 1.49·19-s − 0.0825·20-s + 0.691·21-s − 1.45·23-s + 1.62·24-s + 0.200·25-s + 0.250·26-s − 2.47·27-s − 0.0697·28-s − 0.605·29-s + 0.890·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5910868431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5910868431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 + 3.17T + 3T^{2} \) |
| 13 | \( 1 - 0.829T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.199T + 41T^{2} \) |
| 43 | \( 1 - 0.447T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + 7.95T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 + 0.581T + 89T^{2} \) |
| 97 | \( 1 + 2.15T + 97T^{2} \) |
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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
−8.206445388157879649190675891018, −7.35163999689947472816983581396, −6.46962345928757598138100263632, −6.00367905932793730938510470249, −5.54283579344938775895053174540, −4.61549068069850352301567736021, −4.14768660553636209488763541212, −3.41096912452451613590810967819, −1.91754244481092117653844065177, −0.39277027140333307711002095730,
0.39277027140333307711002095730, 1.91754244481092117653844065177, 3.41096912452451613590810967819, 4.14768660553636209488763541212, 4.61549068069850352301567736021, 5.54283579344938775895053174540, 6.00367905932793730938510470249, 6.46962345928757598138100263632, 7.35163999689947472816983581396, 8.206445388157879649190675891018
