| L(s) = 1 | + 0.542i·3-s + i·5-s + 2.97·7-s + 2.70·9-s − 0.298·11-s − 4.59·13-s − 0.542·15-s − 4.67i·17-s + 1.53·19-s + 1.61i·21-s + (−0.375 + 4.78i)23-s − 25-s + 3.09i·27-s + 9.66·29-s − 4.88i·31-s + ⋯ |
| L(s) = 1 | + 0.313i·3-s + 0.447i·5-s + 1.12·7-s + 0.901·9-s − 0.0898·11-s − 1.27·13-s − 0.140·15-s − 1.13i·17-s + 0.351·19-s + 0.352i·21-s + (−0.0782 + 0.996i)23-s − 0.200·25-s + 0.595i·27-s + 1.79·29-s − 0.876i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.314289799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.314289799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (0.375 - 4.78i)T \) |
| good | 3 | \( 1 - 0.542iT - 3T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 + 0.298T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 + 4.67iT - 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 29 | \( 1 - 9.66T + 29T^{2} \) |
| 31 | \( 1 + 4.88iT - 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 9.97iT - 47T^{2} \) |
| 53 | \( 1 - 5.22iT - 53T^{2} \) |
| 59 | \( 1 + 2.30iT - 59T^{2} \) |
| 61 | \( 1 - 4.22iT - 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 1.21iT - 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 1.00iT - 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−8.559076368895944179607228194141, −7.64537523966263033846466075558, −7.36148670729819835023330107938, −6.52557708924683297785966476709, −5.34827429024125869381825989712, −4.81982349508781695741921817178, −4.19034767325100673929064288841, −3.00655672884419668885529173503, −2.19218794029892752235758867281, −1.01892331146828988192140111893,
0.859808383193654898503448903772, 1.80922217938455736257419243900, 2.64097080282977211001403071143, 4.10526116663102666282709417789, 4.60802115263140249150096328895, 5.28545165608354995430382305736, 6.28369591881298308163520945559, 7.08881547480866278989196912333, 7.80599924435417357044697112039, 8.257856167763827375293017511047
