| L(s) = 1 | − 1.60·2-s + 0.573·4-s + 1.49·5-s + 2.64·7-s + 2.28·8-s − 2.39·10-s − 0.452·11-s − 4.89·13-s − 4.24·14-s − 4.81·16-s − 1.33·17-s + 4.58·19-s + 0.855·20-s + 0.725·22-s − 23-s − 2.77·25-s + 7.85·26-s + 1.51·28-s − 29-s + 3.84·31-s + 3.15·32-s + 2.14·34-s + 3.94·35-s + 5.27·37-s − 7.35·38-s + 3.41·40-s + 2.68·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.286·4-s + 0.666·5-s + 0.999·7-s + 0.808·8-s − 0.756·10-s − 0.136·11-s − 1.35·13-s − 1.13·14-s − 1.20·16-s − 0.324·17-s + 1.05·19-s + 0.191·20-s + 0.154·22-s − 0.208·23-s − 0.555·25-s + 1.54·26-s + 0.286·28-s − 0.185·29-s + 0.691·31-s + 0.557·32-s + 0.367·34-s + 0.666·35-s + 0.866·37-s − 1.19·38-s + 0.539·40-s + 0.419·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 0.452T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 5.99T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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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
−7.85123617824902068790201771089, −7.33919899044623332835514166441, −6.52198534642982045869408352740, −5.47640117668056547956945201128, −4.90448579788295306924270896460, −4.26393517874347694533999252633, −2.87579251279399624862338916043, −1.98699170932495344719794580320, −1.31109211349890343756070385349, 0,
1.31109211349890343756070385349, 1.98699170932495344719794580320, 2.87579251279399624862338916043, 4.26393517874347694533999252633, 4.90448579788295306924270896460, 5.47640117668056547956945201128, 6.52198534642982045869408352740, 7.33919899044623332835514166441, 7.85123617824902068790201771089
