L(s) = 1 | + 3-s − 0.563·5-s − 3.34·7-s + 9-s − 6.11·11-s + 2.34·13-s − 0.563·15-s + 6.16·17-s + 4.83·19-s − 3.34·21-s + 2.50·23-s − 4.68·25-s + 27-s − 5.48·29-s + 7.48·31-s − 6.11·33-s + 1.88·35-s + 0.750·37-s + 2.34·39-s + 4.56·41-s − 6.66·43-s − 0.563·45-s + 7.06·47-s + 4.19·49-s + 6.16·51-s + 5.57·53-s + 3.44·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.251·5-s − 1.26·7-s + 0.333·9-s − 1.84·11-s + 0.650·13-s − 0.145·15-s + 1.49·17-s + 1.11·19-s − 0.730·21-s + 0.521·23-s − 0.936·25-s + 0.192·27-s − 1.01·29-s + 1.34·31-s − 1.06·33-s + 0.318·35-s + 0.123·37-s + 0.375·39-s + 0.713·41-s − 1.01·43-s − 0.0839·45-s + 1.03·47-s + 0.599·49-s + 0.863·51-s + 0.765·53-s + 0.464·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 5 | \( 1 + 0.563T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 - 0.750T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 - 7.06T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−7.55392974581229084729837546918, −7.42691134350293803363741735057, −6.14487610274822972722738167318, −5.69726400107843410811153381499, −4.84753953938707033614517014402, −3.77144802648178336628732967241, −3.09527062061211385190426130275, −2.70971572037674494161417778774, −1.28387246677269288720817225187, 0,
1.28387246677269288720817225187, 2.70971572037674494161417778774, 3.09527062061211385190426130275, 3.77144802648178336628732967241, 4.84753953938707033614517014402, 5.69726400107843410811153381499, 6.14487610274822972722738167318, 7.42691134350293803363741735057, 7.55392974581229084729837546918