| L(s) = 1 | + 3-s + 3.16·5-s + 1.16·7-s + 9-s + 4·13-s + 3.16·15-s + 0.837·17-s − 5.16·19-s + 1.16·21-s + 23-s + 5.00·25-s + 27-s − 4.32·29-s − 6.32·31-s + 3.67·35-s + 4.32·37-s + 4·39-s − 2·41-s + 5.16·43-s + 3.16·45-s + 12.3·47-s − 5.64·49-s + 0.837·51-s − 13.4·53-s − 5.16·57-s − 4.32·59-s − 12.3·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.41·5-s + 0.439·7-s + 0.333·9-s + 1.10·13-s + 0.816·15-s + 0.203·17-s − 1.18·19-s + 0.253·21-s + 0.208·23-s + 1.00·25-s + 0.192·27-s − 0.803·29-s − 1.13·31-s + 0.621·35-s + 0.710·37-s + 0.640·39-s − 0.312·41-s + 0.787·43-s + 0.471·45-s + 1.79·47-s − 0.807·49-s + 0.117·51-s − 1.85·53-s − 0.683·57-s − 0.563·59-s − 1.57·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.725625886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725625886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 0.837T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.637109566619786988673654716777, −9.125061439601864799698190805902, −8.360500725723851675291775276462, −7.43243420797839236200318878433, −6.30052912821442209587636101703, −5.77063189199863048659923674489, −4.64486338868454207255687840136, −3.53944822494754133565740177125, −2.29138431559434990321478437130, −1.48552685582086064159647029437,
1.48552685582086064159647029437, 2.29138431559434990321478437130, 3.53944822494754133565740177125, 4.64486338868454207255687840136, 5.77063189199863048659923674489, 6.30052912821442209587636101703, 7.43243420797839236200318878433, 8.360500725723851675291775276462, 9.125061439601864799698190805902, 9.637109566619786988673654716777
