| L(s) = 1 | − 1.35·3-s + 4-s + 1.89·5-s + 0.834·9-s − 1.97·11-s − 1.35·12-s + 0.490·13-s − 2.56·15-s + 16-s + 1.09·19-s + 1.89·20-s + 2.57·25-s + 0.223·27-s + 2.67·33-s + 0.834·36-s − 0.665·39-s − 0.803·41-s − 1.97·44-s + 1.57·45-s − 1.35·48-s + 49-s + 0.490·52-s − 1.75·53-s − 3.73·55-s − 1.48·57-s + 0.490·59-s − 2.56·60-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 4-s + 1.89·5-s + 0.834·9-s − 1.97·11-s − 1.35·12-s + 0.490·13-s − 2.56·15-s + 16-s + 1.09·19-s + 1.89·20-s + 2.57·25-s + 0.223·27-s + 2.67·33-s + 0.834·36-s − 0.665·39-s − 0.803·41-s − 1.97·44-s + 1.57·45-s − 1.35·48-s + 49-s + 0.490·52-s − 1.75·53-s − 3.73·55-s − 1.48·57-s + 0.490·59-s − 2.56·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.291863439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291863439\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2339 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.35T + T^{2} \) |
| 5 | \( 1 - 1.89T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.97T + T^{2} \) |
| 13 | \( 1 - 0.490T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.09T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.803T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.75T + T^{2} \) |
| 59 | \( 1 - 0.490T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.803T + T^{2} \) |
| 71 | \( 1 - 1.57T + T^{2} \) |
| 73 | \( 1 + 0.165T + T^{2} \) |
| 79 | \( 1 + 0.803T + T^{2} \) |
| 83 | \( 1 - 1.09T + T^{2} \) |
| 89 | \( 1 - 1.57T + T^{2} \) |
| 97 | \( 1 + 1.75T + T^{2} \) |
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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
−9.543377257845747041852345806300, −8.319283527898779628993505254766, −7.37228941889148864627697512395, −6.56691229593160049261815525494, −5.97162473004084557424582101022, −5.37622729207927917025290815077, −5.04284950175733825309964434350, −3.07462780715204783813260831203, −2.30897775229814230531731645382, −1.26066619482374736120787607549,
1.26066619482374736120787607549, 2.30897775229814230531731645382, 3.07462780715204783813260831203, 5.04284950175733825309964434350, 5.37622729207927917025290815077, 5.97162473004084557424582101022, 6.56691229593160049261815525494, 7.37228941889148864627697512395, 8.319283527898779628993505254766, 9.543377257845747041852345806300
