| L(s) = 1 | − 3-s + 2.47·5-s + 7-s + 9-s + 5.95·11-s − 0.137·13-s − 2.47·15-s + 4.61·17-s + 19-s − 21-s + 23-s + 1.13·25-s − 27-s − 1.65·29-s − 2.61·31-s − 5.95·33-s + 2.47·35-s − 11.5·37-s + 0.137·39-s − 3.68·41-s + 11.0·43-s + 2.47·45-s + 5.34·47-s + 49-s − 4.61·51-s − 5.13·53-s + 14.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.10·5-s + 0.377·7-s + 0.333·9-s + 1.79·11-s − 0.0380·13-s − 0.639·15-s + 1.11·17-s + 0.229·19-s − 0.218·21-s + 0.208·23-s + 0.227·25-s − 0.192·27-s − 0.308·29-s − 0.469·31-s − 1.03·33-s + 0.418·35-s − 1.90·37-s + 0.0219·39-s − 0.574·41-s + 1.69·43-s + 0.369·45-s + 0.778·47-s + 0.142·49-s − 0.646·51-s − 0.705·53-s + 1.98·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179589314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179589314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.47T + 5T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + 0.137T + 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 - 4.52T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.340T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 + 2.88T + 97T^{2} \) |
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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
−9.292349870942084348864024404797, −8.630373144098292734605416833654, −7.40455283104845229980251909802, −6.76643453071752925749655738682, −5.85305523640888769939048075958, −5.44332375842507692966498368726, −4.32690802478894378262859556433, −3.39944760375946459686698741226, −1.91633158534089168971005849212, −1.15106656752182189942094485926,
1.15106656752182189942094485926, 1.91633158534089168971005849212, 3.39944760375946459686698741226, 4.32690802478894378262859556433, 5.44332375842507692966498368726, 5.85305523640888769939048075958, 6.76643453071752925749655738682, 7.40455283104845229980251909802, 8.630373144098292734605416833654, 9.292349870942084348864024404797
