| L(s) = 1 | − 1.41·3-s − 2.82·5-s − 0.999·9-s − 6·11-s + 5.65·13-s + 4.00·15-s + 1.41·17-s + 4.24·19-s + 4·23-s + 3.00·25-s + 5.65·27-s + 6·29-s + 2.82·31-s + 8.48·33-s − 2·37-s − 8.00·39-s − 1.41·41-s − 10·43-s + 2.82·45-s − 2.82·47-s − 2.00·51-s + 2·53-s + 16.9·55-s − 6·57-s − 1.41·59-s + 8.48·61-s − 16.0·65-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.26·5-s − 0.333·9-s − 1.80·11-s + 1.56·13-s + 1.03·15-s + 0.342·17-s + 0.973·19-s + 0.834·23-s + 0.600·25-s + 1.08·27-s + 1.11·29-s + 0.508·31-s + 1.47·33-s − 0.328·37-s − 1.28·39-s − 0.220·41-s − 1.52·43-s + 0.421·45-s − 0.412·47-s − 0.280·51-s + 0.274·53-s + 2.28·55-s − 0.794·57-s − 0.184·59-s + 1.08·61-s − 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.326162772601717295568240502081, −7.62908365377856743192213121384, −6.83621911996173422599085624556, −5.92274649529308365329734314820, −5.23945503338918113080317033238, −4.59479565183234145611870578811, −3.41516722506964026754044032182, −2.89507042993784278524727808469, −1.08720219941479065756890632079, 0,
1.08720219941479065756890632079, 2.89507042993784278524727808469, 3.41516722506964026754044032182, 4.59479565183234145611870578811, 5.23945503338918113080317033238, 5.92274649529308365329734314820, 6.83621911996173422599085624556, 7.62908365377856743192213121384, 8.326162772601717295568240502081
