| L(s) = 1 | + 2-s + 4-s − 3.64·5-s + 8-s − 3.64·10-s − 11-s + 5·13-s + 16-s − 6·17-s − 0.354·19-s − 3.64·20-s − 22-s + 3.64·23-s + 8.29·25-s + 5·26-s + 4.29·29-s − 4·31-s + 32-s − 6·34-s − 1.64·37-s − 0.354·38-s − 3.64·40-s + 4.93·41-s − 4·43-s − 44-s + 3.64·46-s − 13.2·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.63·5-s + 0.353·8-s − 1.15·10-s − 0.301·11-s + 1.38·13-s + 0.250·16-s − 1.45·17-s − 0.0812·19-s − 0.815·20-s − 0.213·22-s + 0.760·23-s + 1.65·25-s + 0.980·26-s + 0.796·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.270·37-s − 0.0574·38-s − 0.576·40-s + 0.771·41-s − 0.609·43-s − 0.150·44-s + 0.537·46-s − 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 0.354T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 - 0.645T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−7.22309125229382672097898345309, −6.68816731188156386280414541199, −6.05498629948923210132066374062, −5.00073920335338608391937754295, −4.53724437853616671412808749454, −3.76058789494987738579122629934, −3.35755832984024996535635168196, −2.40333715280629236486415429709, −1.20013032064705326129062320203, 0,
1.20013032064705326129062320203, 2.40333715280629236486415429709, 3.35755832984024996535635168196, 3.76058789494987738579122629934, 4.53724437853616671412808749454, 5.00073920335338608391937754295, 6.05498629948923210132066374062, 6.68816731188156386280414541199, 7.22309125229382672097898345309
