| L(s) = 1 | − 2-s + 3-s + 4-s + 0.937·5-s − 6-s + 7-s − 8-s + 9-s − 0.937·10-s − 0.541·11-s + 12-s − 14-s + 0.937·15-s + 16-s − 6.88·17-s − 18-s + 1.41·19-s + 0.937·20-s + 21-s + 0.541·22-s + 3.20·23-s − 24-s − 4.12·25-s + 27-s + 28-s − 0.718·29-s − 0.937·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.419·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.296·10-s − 0.163·11-s + 0.288·12-s − 0.267·14-s + 0.242·15-s + 0.250·16-s − 1.66·17-s − 0.235·18-s + 0.325·19-s + 0.209·20-s + 0.218·21-s + 0.115·22-s + 0.667·23-s − 0.204·24-s − 0.824·25-s + 0.192·27-s + 0.188·28-s − 0.133·29-s − 0.171·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.937T + 5T^{2} \) |
| 11 | \( 1 + 0.541T + 11T^{2} \) |
| 17 | \( 1 + 6.88T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 + 0.718T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 + 0.659T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.58782907306681017532818832178, −7.11233758789436758847589003777, −6.36133418643246993240605205921, −5.54971430980364171361609353622, −4.72391333687900278100071823551, −3.87658783555335638932932109411, −2.90877018707212966581757102270, −2.12362756351627301781177945616, −1.46321689966153493921957256908, 0,
1.46321689966153493921957256908, 2.12362756351627301781177945616, 2.90877018707212966581757102270, 3.87658783555335638932932109411, 4.72391333687900278100071823551, 5.54971430980364171361609353622, 6.36133418643246993240605205921, 7.11233758789436758847589003777, 7.58782907306681017532818832178
