| L(s) = 1 | + 1.37·2-s − 3-s − 0.107·4-s − 1.37·6-s + 1.29·7-s − 2.89·8-s + 9-s − 5.18·11-s + 0.107·12-s + 1.56·13-s + 1.77·14-s − 3.77·16-s + 5.67·17-s + 1.37·18-s + 0.785·19-s − 1.29·21-s − 7.13·22-s − 9.21·23-s + 2.89·24-s + 2.14·26-s − 27-s − 0.138·28-s + 9.36·29-s + 9.91·31-s + 0.605·32-s + 5.18·33-s + 7.80·34-s + ⋯ |
| L(s) = 1 | + 0.972·2-s − 0.577·3-s − 0.0536·4-s − 0.561·6-s + 0.488·7-s − 1.02·8-s + 0.333·9-s − 1.56·11-s + 0.0309·12-s + 0.432·13-s + 0.475·14-s − 0.943·16-s + 1.37·17-s + 0.324·18-s + 0.180·19-s − 0.282·21-s − 1.52·22-s − 1.92·23-s + 0.591·24-s + 0.421·26-s − 0.192·27-s − 0.0261·28-s + 1.73·29-s + 1.78·31-s + 0.107·32-s + 0.902·33-s + 1.33·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 - 0.785T + 19T^{2} \) |
| 23 | \( 1 + 9.21T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 - 9.91T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 + 0.756T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 - 4.30T + 47T^{2} \) |
| 53 | \( 1 + 6.56T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 + 0.507T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.54101044400449758639829299797, −6.40830696563117175275965642169, −5.93275844566820884237769741675, −5.31166158509670005416583091764, −4.75283382174591048361302284279, −4.14222333072242422666619276908, −3.17881690028802648298964048158, −2.53437371244969542919683624734, −1.21829159887313926369769587316, 0,
1.21829159887313926369769587316, 2.53437371244969542919683624734, 3.17881690028802648298964048158, 4.14222333072242422666619276908, 4.75283382174591048361302284279, 5.31166158509670005416583091764, 5.93275844566820884237769741675, 6.40830696563117175275965642169, 7.54101044400449758639829299797
