| L(s) = 1 | + 2.57·2-s − 3-s + 4.63·4-s − 5-s − 2.57·6-s + 6.78·8-s + 9-s − 2.57·10-s − 0.941·11-s − 4.63·12-s − 0.424·13-s + 15-s + 8.21·16-s − 7.84·17-s + 2.57·18-s − 0.634·19-s − 4.63·20-s − 2.42·22-s − 2.69·23-s − 6.78·24-s − 4·25-s − 1.09·26-s − 27-s − 1.51·29-s + 2.57·30-s − 4.09·31-s + 7.57·32-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.577·3-s + 2.31·4-s − 0.447·5-s − 1.05·6-s + 2.39·8-s + 0.333·9-s − 0.814·10-s − 0.283·11-s − 1.33·12-s − 0.117·13-s + 0.258·15-s + 2.05·16-s − 1.90·17-s + 0.607·18-s − 0.145·19-s − 1.03·20-s − 0.516·22-s − 0.561·23-s − 1.38·24-s − 0.800·25-s − 0.214·26-s − 0.192·27-s − 0.281·29-s + 0.470·30-s − 0.735·31-s + 1.33·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 0.941T + 11T^{2} \) |
| 13 | \( 1 + 0.424T + 13T^{2} \) |
| 17 | \( 1 + 7.84T + 17T^{2} \) |
| 19 | \( 1 + 0.634T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 + 0.210T + 53T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 61 | \( 1 - 6.36T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 0.579T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 + 7.94T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.34246735386532515092116086993, −6.76076598911967721284958120471, −6.08936527431585330371686435522, −5.53915224568946106378075041863, −4.63883558061033312245526983166, −4.28268158928617262950387133128, −3.53566182963127758508633552894, −2.52059176203763092995664483926, −1.80155731051266448009977759617, 0,
1.80155731051266448009977759617, 2.52059176203763092995664483926, 3.53566182963127758508633552894, 4.28268158928617262950387133128, 4.63883558061033312245526983166, 5.53915224568946106378075041863, 6.08936527431585330371686435522, 6.76076598911967721284958120471, 7.34246735386532515092116086993
