| L(s) = 1 | + 3-s − 2·4-s − 2.44·5-s + 0.449·7-s + 9-s − 2·12-s + 13-s − 2.44·15-s + 4·16-s + 3.44·17-s − 19-s + 4.89·20-s + 0.449·21-s − 23-s + 0.999·25-s + 27-s − 0.898·28-s + 29-s − 5.55·31-s − 1.10·35-s − 2·36-s + 0.101·37-s + 39-s + 5.34·41-s − 9.89·43-s − 2.44·45-s − 7.55·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.09·5-s + 0.169·7-s + 0.333·9-s − 0.577·12-s + 0.277·13-s − 0.632·15-s + 16-s + 0.836·17-s − 0.229·19-s + 1.09·20-s + 0.0980·21-s − 0.208·23-s + 0.199·25-s + 0.192·27-s − 0.169·28-s + 0.185·29-s − 0.996·31-s − 0.186·35-s − 0.333·36-s + 0.0166·37-s + 0.160·39-s + 0.835·41-s − 1.50·43-s − 0.365·45-s − 1.10·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 0.101T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 0.898T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3.89T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 0.550T + 89T^{2} \) |
| 97 | \( 1 - 7.10T + 97T^{2} \) |
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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
−8.675678596954291857478276909862, −7.956326453341084321704128478652, −7.65654790965138032112514291698, −6.45808048593907849488503444264, −5.33961672601415239137363710270, −4.51904068327314592633435541565, −3.75858411052598056594396779379, −3.13760818701733725459935368523, −1.48482094009180195888644385117, 0,
1.48482094009180195888644385117, 3.13760818701733725459935368523, 3.75858411052598056594396779379, 4.51904068327314592633435541565, 5.33961672601415239137363710270, 6.45808048593907849488503444264, 7.65654790965138032112514291698, 7.956326453341084321704128478652, 8.675678596954291857478276909862
