| L(s) = 1 | − 1.41·5-s − 2.41·7-s + 11-s + 0.414·13-s + 2.24·17-s + 3.24·19-s + 5.65·23-s − 2.99·25-s − 3.41·29-s − 7.65·31-s + 3.41·35-s + 1.82·37-s + 2.82·41-s − 0.343·43-s − 0.242·47-s − 1.17·49-s − 13.8·53-s − 1.41·55-s + 5.41·59-s − 10.4·61-s − 0.585·65-s − 14.3·67-s + 4·71-s − 8.89·73-s − 2.41·77-s − 14.0·79-s − 1.65·83-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 0.912·7-s + 0.301·11-s + 0.114·13-s + 0.543·17-s + 0.743·19-s + 1.17·23-s − 0.599·25-s − 0.634·29-s − 1.37·31-s + 0.577·35-s + 0.300·37-s + 0.441·41-s − 0.0523·43-s − 0.0353·47-s − 0.167·49-s − 1.90·53-s − 0.190·55-s + 0.704·59-s − 1.33·61-s − 0.0726·65-s − 1.74·67-s + 0.474·71-s − 1.04·73-s − 0.275·77-s − 1.58·79-s − 0.181·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + 0.242T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−8.681237266599858343077904352755, −7.60351995057023081646778173472, −7.24972432662174255385677075924, −6.23381859710156451987175527471, −5.53205385150159022033078492249, −4.48400780727700478182195133345, −3.53676584629914865865578372000, −2.99134373761070152248490499683, −1.45405574377872557747187992338, 0,
1.45405574377872557747187992338, 2.99134373761070152248490499683, 3.53676584629914865865578372000, 4.48400780727700478182195133345, 5.53205385150159022033078492249, 6.23381859710156451987175527471, 7.24972432662174255385677075924, 7.60351995057023081646778173472, 8.681237266599858343077904352755
