| L(s) = 1 | − 2-s + 3-s + 4-s − 1.56·5-s − 6-s + 1.95·7-s − 8-s + 9-s + 1.56·10-s − 1.08·11-s + 12-s − 0.218·13-s − 1.95·14-s − 1.56·15-s + 16-s + 7.65·17-s − 18-s − 7.93·19-s − 1.56·20-s + 1.95·21-s + 1.08·22-s + 23-s − 24-s − 2.56·25-s + 0.218·26-s + 27-s + 1.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s + 0.738·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s − 0.325·11-s + 0.288·12-s − 0.0605·13-s − 0.521·14-s − 0.403·15-s + 0.250·16-s + 1.85·17-s − 0.235·18-s − 1.82·19-s − 0.349·20-s + 0.426·21-s + 0.230·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s + 0.0428·26-s + 0.192·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.562789605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562789605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 + 0.218T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 + 0.314T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 1.34T + 43T^{2} \) |
| 47 | \( 1 - 6.23T + 47T^{2} \) |
| 53 | \( 1 + 8.75T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 5.49T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 + 9.40T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 4.40T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.356157007317774227264774970488, −7.80743435342889869479355278626, −7.43565541729463868194209663802, −6.39007994534109148126914242761, −5.54247687550545636234077160333, −4.51447147742858893683749105857, −3.79363567087771109781251529602, −2.80978638299004530061618342195, −1.92491183332700487710984118619, −0.78386415290010982350728773879,
0.78386415290010982350728773879, 1.92491183332700487710984118619, 2.80978638299004530061618342195, 3.79363567087771109781251529602, 4.51447147742858893683749105857, 5.54247687550545636234077160333, 6.39007994534109148126914242761, 7.43565541729463868194209663802, 7.80743435342889869479355278626, 8.356157007317774227264774970488
