| L(s) = 1 | + 3.46·5-s − 24.2·7-s + 48·11-s − 41.5·13-s − 54·17-s + 4·19-s + 173.·23-s − 113·25-s + 162.·29-s + 58.8·31-s − 84·35-s + 325.·37-s − 294·41-s − 188·43-s − 505.·47-s + 245·49-s + 744.·53-s + 166.·55-s + 252·59-s + 90.0·61-s − 144·65-s − 628·67-s − 6.92·71-s + 1.00e3·73-s − 1.16e3·77-s − 1.34e3·79-s − 720·83-s + ⋯ |
| L(s) = 1 | + 0.309·5-s − 1.30·7-s + 1.31·11-s − 0.886·13-s − 0.770·17-s + 0.0482·19-s + 1.57·23-s − 0.904·25-s + 1.04·29-s + 0.341·31-s − 0.405·35-s + 1.44·37-s − 1.11·41-s − 0.666·43-s − 1.56·47-s + 0.714·49-s + 1.93·53-s + 0.407·55-s + 0.556·59-s + 0.189·61-s − 0.274·65-s − 1.14·67-s − 0.0115·71-s + 1.61·73-s − 1.72·77-s − 1.90·79-s − 0.952·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 - 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 294T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188T + 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 252T + 2.05e5T^{2} \) |
| 61 | \( 1 - 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 628T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−8.478388700401409584633291309844, −7.22546943586033559283964107687, −6.67511573487922455440721673141, −6.16171955983715568022898361792, −5.06246514340347868467990811307, −4.18801816091909253913323969325, −3.24839669888263582787658621778, −2.44008583927924792198554814079, −1.17194468480189692309170278009, 0,
1.17194468480189692309170278009, 2.44008583927924792198554814079, 3.24839669888263582787658621778, 4.18801816091909253913323969325, 5.06246514340347868467990811307, 6.16171955983715568022898361792, 6.67511573487922455440721673141, 7.22546943586033559283964107687, 8.478388700401409584633291309844
