| L(s) = 1 | + 2.32·3-s − 4.00·5-s + 7-s + 2.38·9-s + 11-s + 13-s − 9.29·15-s − 2.56·17-s − 0.392·19-s + 2.32·21-s + 4.91·23-s + 11.0·25-s − 1.42·27-s − 10.5·29-s + 3.48·31-s + 2.32·33-s − 4.00·35-s + 11.2·37-s + 2.32·39-s − 9.26·41-s − 10.8·43-s − 9.55·45-s − 4.98·47-s + 49-s − 5.94·51-s + 3.32·53-s − 4.00·55-s + ⋯ |
| L(s) = 1 | + 1.33·3-s − 1.79·5-s + 0.377·7-s + 0.795·9-s + 0.301·11-s + 0.277·13-s − 2.40·15-s − 0.621·17-s − 0.0900·19-s + 0.506·21-s + 1.02·23-s + 2.21·25-s − 0.274·27-s − 1.95·29-s + 0.625·31-s + 0.403·33-s − 0.677·35-s + 1.85·37-s + 0.371·39-s − 1.44·41-s − 1.66·43-s − 1.42·45-s − 0.726·47-s + 0.142·49-s − 0.832·51-s + 0.456·53-s − 0.540·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 0.392T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 3.48T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 1.98T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 - 9.91T + 97T^{2} \) |
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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
−7.74163922728931226210210454651, −7.09986600248953076738628850826, −6.36595325604733556686358545642, −5.08082251015545344580750124323, −4.43972221801943144473431772655, −3.68744651225800557902920628007, −3.32916375528002342751741831019, −2.41871953741987757502024976059, −1.35103875534390519695266869762, 0,
1.35103875534390519695266869762, 2.41871953741987757502024976059, 3.32916375528002342751741831019, 3.68744651225800557902920628007, 4.43972221801943144473431772655, 5.08082251015545344580750124323, 6.36595325604733556686358545642, 7.09986600248953076738628850826, 7.74163922728931226210210454651
