| L(s) = 1 | + 0.208·2-s − 1.95·4-s + 2.17·5-s − 2.92·7-s − 0.826·8-s + 0.453·10-s + 4.59·11-s − 4.31·13-s − 0.611·14-s + 3.74·16-s + 1.26·17-s + 2.45·19-s − 4.25·20-s + 0.960·22-s − 23-s − 0.279·25-s − 0.901·26-s + 5.72·28-s + 29-s − 9.32·31-s + 2.43·32-s + 0.263·34-s − 6.36·35-s + 0.827·37-s + 0.513·38-s − 1.79·40-s − 1.17·41-s + ⋯ |
| L(s) = 1 | + 0.147·2-s − 0.978·4-s + 0.971·5-s − 1.10·7-s − 0.292·8-s + 0.143·10-s + 1.38·11-s − 1.19·13-s − 0.163·14-s + 0.935·16-s + 0.306·17-s + 0.563·19-s − 0.950·20-s + 0.204·22-s − 0.208·23-s − 0.0558·25-s − 0.176·26-s + 1.08·28-s + 0.185·29-s − 1.67·31-s + 0.430·32-s + 0.0452·34-s − 1.07·35-s + 0.136·37-s + 0.0832·38-s − 0.283·40-s − 0.183·41-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 | \( 1 - 0.208T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 - 0.827T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 2.72T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 + 6.56T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 0.738T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.66878018385652145581293354176, −6.88889052019972646960784568298, −6.22983608898027812006887260132, −5.56152870244914991443099330799, −4.94732093189442849132868184473, −3.92861305844655167529671599578, −3.41461791575467620446390896659, −2.36941733291064448306234389121, −1.26731681958339421977207825204, 0,
1.26731681958339421977207825204, 2.36941733291064448306234389121, 3.41461791575467620446390896659, 3.92861305844655167529671599578, 4.94732093189442849132868184473, 5.56152870244914991443099330799, 6.22983608898027812006887260132, 6.88889052019972646960784568298, 7.66878018385652145581293354176
