| L(s) = 1 | + (−1.61 + 0.618i)3-s + 1.54i·5-s + 2.42i·7-s + (2.23 − 2.00i)9-s + 2.73·11-s + 5.46·13-s + (−0.956 − 2.50i)15-s − 4.19i·17-s − 1.54i·19-s + (−1.49 − 3.91i)21-s − 23-s + 2.60·25-s + (−2.38 + 4.61i)27-s + 4.47i·29-s + 5.60i·31-s + ⋯ |
| L(s) = 1 | + (−0.934 + 0.356i)3-s + 0.692i·5-s + 0.914i·7-s + (0.745 − 0.666i)9-s + 0.823·11-s + 1.51·13-s + (−0.246 − 0.646i)15-s − 1.01i·17-s − 0.355i·19-s + (−0.326 − 0.854i)21-s − 0.208·23-s + 0.521·25-s + (−0.458 + 0.888i)27-s + 0.830i·29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300804377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300804377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.54iT - 5T^{2} \) |
| 7 | \( 1 - 2.42iT - 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 4.19iT - 17T^{2} \) |
| 19 | \( 1 + 1.54iT - 19T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 5.60iT - 31T^{2} \) |
| 37 | \( 1 + 0.872T + 37T^{2} \) |
| 41 | \( 1 + 4.22iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 0.539iT - 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 - 13.2iT - 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.21659493715166151166910162851, −9.109399036517172350852158307641, −8.742883129426196590195845158546, −7.22609607462366948670039546461, −6.56588504054099859165577389178, −5.86169215640799209490659880741, −5.00429114147677911519142991198, −3.87899283820801250822813035233, −2.90335479495868081065496431158, −1.25542334986011029906005460369,
0.806943432712877491278731418902, 1.68030860040778531839612844561, 3.82167825400971151238559482347, 4.29838564478953950685003738827, 5.55819414442933615770243874725, 6.25681908031082782303705909671, 6.99029776194431335764456471356, 8.062274214385581815147150001496, 8.687649090740798856596552457249, 9.874554295435290537810474045543
