| L(s) = 1 | − 2.38·2-s − 3-s + 3.66·4-s − 5-s + 2.38·6-s − 3.58·7-s − 3.96·8-s + 9-s + 2.38·10-s − 3.63·11-s − 3.66·12-s − 2.93·13-s + 8.53·14-s + 15-s + 2.10·16-s + 4.41·17-s − 2.38·18-s + 5.72·19-s − 3.66·20-s + 3.58·21-s + 8.64·22-s + 3.96·24-s + 25-s + 6.99·26-s − 27-s − 13.1·28-s − 1.14·29-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.447·5-s + 0.971·6-s − 1.35·7-s − 1.40·8-s + 0.333·9-s + 0.752·10-s − 1.09·11-s − 1.05·12-s − 0.814·13-s + 2.28·14-s + 0.258·15-s + 0.526·16-s + 1.07·17-s − 0.561·18-s + 1.31·19-s − 0.819·20-s + 0.782·21-s + 1.84·22-s + 0.809·24-s + 0.200·25-s + 1.37·26-s − 0.192·27-s − 2.48·28-s − 0.213·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 5.61T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 9.68T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 5.72T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 - 4.06T + 97T^{2} \) |
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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
−7.67322197799710231607101500477, −7.05753090294792407330159308364, −6.44112935202731177199045582532, −5.52636343175494156538253896372, −4.95503027782634516547963953358, −3.52584924421269847057473312690, −2.99693114528705331903470211085, −1.94704378372626620340137193243, −0.74003282871971393424653332680, 0,
0.74003282871971393424653332680, 1.94704378372626620340137193243, 2.99693114528705331903470211085, 3.52584924421269847057473312690, 4.95503027782634516547963953358, 5.52636343175494156538253896372, 6.44112935202731177199045582532, 7.05753090294792407330159308364, 7.67322197799710231607101500477
