| L(s) = 1 | + 2.82·3-s − 2·4-s − 0.476·5-s + 3.82·7-s + 4.95·9-s − 2.29·11-s − 5.64·12-s + 13-s − 1.34·15-s + 4·16-s − 0.952·17-s + 4.29·19-s + 0.952·20-s + 10.7·21-s − 23-s − 4.77·25-s + 5.50·27-s − 7.64·28-s − 4.59·29-s − 4·31-s − 6.47·33-s − 1.82·35-s − 9.90·36-s + 1.13·37-s + 2.82·39-s − 2.68·41-s − 6.68·43-s + ⋯ |
| L(s) = 1 | + 1.62·3-s − 4-s − 0.213·5-s + 1.44·7-s + 1.65·9-s − 0.692·11-s − 1.62·12-s + 0.277·13-s − 0.346·15-s + 16-s − 0.231·17-s + 0.985·19-s + 0.213·20-s + 2.35·21-s − 0.208·23-s − 0.954·25-s + 1.05·27-s − 1.44·28-s − 0.852·29-s − 0.718·31-s − 1.12·33-s − 0.307·35-s − 1.65·36-s + 0.186·37-s + 0.451·39-s − 0.419·41-s − 1.01·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868685269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868685269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.476T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 17 | \( 1 + 0.952T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 + 6.68T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 5.04T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 2.59T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 0.210T + 97T^{2} \) |
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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
−11.81926791253716497252016418125, −10.63063704457279138576852728384, −9.528744652286549927419925131719, −8.826169559707782152522843065018, −7.917277826605131831984888069036, −7.64339432618711629793334564189, −5.41516774724452780586831507659, −4.37755958592896281665451562997, −3.36638437664250316431869159781, −1.80508457107762701099895561645,
1.80508457107762701099895561645, 3.36638437664250316431869159781, 4.37755958592896281665451562997, 5.41516774724452780586831507659, 7.64339432618711629793334564189, 7.917277826605131831984888069036, 8.826169559707782152522843065018, 9.528744652286549927419925131719, 10.63063704457279138576852728384, 11.81926791253716497252016418125
