| L(s) = 1 | − 3.41·3-s + 5-s + 0.828·7-s + 8.65·9-s − 2·11-s − 6.24·13-s − 3.41·15-s + 0.828·17-s − 19-s − 2.82·21-s − 6·23-s + 25-s − 19.3·27-s − 6.48·29-s − 6.82·31-s + 6.82·33-s + 0.828·35-s − 1.75·37-s + 21.3·39-s + 3.65·41-s + 4.82·43-s + 8.65·45-s − 4.82·47-s − 6.31·49-s − 2.82·51-s + 9.07·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 1.97·3-s + 0.447·5-s + 0.313·7-s + 2.88·9-s − 0.603·11-s − 1.73·13-s − 0.881·15-s + 0.200·17-s − 0.229·19-s − 0.617·21-s − 1.25·23-s + 0.200·25-s − 3.71·27-s − 1.20·29-s − 1.22·31-s + 1.18·33-s + 0.140·35-s − 0.288·37-s + 3.41·39-s + 0.571·41-s + 0.736·43-s + 1.29·45-s − 0.704·47-s − 0.901·49-s − 0.396·51-s + 1.24·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−10.94256066319059589801682795120, −10.16623922017043054936950871610, −9.537240522241727969188861774921, −7.69325377850132072651736083877, −6.99267862041749681055782655233, −5.78573202136722975021606241343, −5.26924804531103241711560797133, −4.27638772863881814162798045888, −1.95569357302930458987582739256, 0,
1.95569357302930458987582739256, 4.27638772863881814162798045888, 5.26924804531103241711560797133, 5.78573202136722975021606241343, 6.99267862041749681055782655233, 7.69325377850132072651736083877, 9.537240522241727969188861774921, 10.16623922017043054936950871610, 10.94256066319059589801682795120
