| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 11-s − 13-s + 2·15-s + 5·19-s + 21-s − 4·23-s − 25-s − 27-s − 5·31-s + 33-s + 2·35-s − 7·37-s + 39-s − 6·41-s − 43-s − 2·45-s + 2·47-s − 6·49-s + 14·53-s + 2·55-s − 5·57-s − 14·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s + 1.14·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.898·31-s + 0.174·33-s + 0.338·35-s − 1.15·37-s + 0.160·39-s − 0.937·41-s − 0.152·43-s − 0.298·45-s + 0.291·47-s − 6/7·49-s + 1.92·53-s + 0.269·55-s − 0.662·57-s − 1.82·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.39120698710974, −13.18782016259745, −12.28082895822157, −12.16099239574490, −11.80359561985064, −11.21226061726197, −10.79245394309736, −10.15906014341814, −9.903034526625026, −9.261549371862716, −8.737668899086595, −8.128638779202438, −7.668126735830737, −7.182412966798404, −6.884456856063950, −6.056680931280105, −5.685022478405804, −5.113795635936492, −4.603330950787701, −3.973197162022702, −3.442538194588137, −3.047184405926706, −2.099262310462860, −1.529791756100903, −0.5728488559369847, 0,
0.5728488559369847, 1.529791756100903, 2.099262310462860, 3.047184405926706, 3.442538194588137, 3.973197162022702, 4.603330950787701, 5.113795635936492, 5.685022478405804, 6.056680931280105, 6.884456856063950, 7.182412966798404, 7.668126735830737, 8.128638779202438, 8.737668899086595, 9.261549371862716, 9.903034526625026, 10.15906014341814, 10.79245394309736, 11.21226061726197, 11.80359561985064, 12.16099239574490, 12.28082895822157, 13.18782016259745, 13.39120698710974
