| L(s) = 1 | + i·2-s + 1.58i·3-s − 4-s + (−2.13 − 0.651i)5-s − 1.58·6-s − i·7-s − i·8-s + 0.499·9-s + (0.651 − 2.13i)10-s + 2.47·11-s − 1.58i·12-s − 1.48i·13-s + 14-s + (1.02 − 3.38i)15-s + 16-s − i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.913i·3-s − 0.5·4-s + (−0.956 − 0.291i)5-s − 0.645·6-s − 0.377i·7-s − 0.353i·8-s + 0.166·9-s + (0.205 − 0.676i)10-s + 0.747·11-s − 0.456i·12-s − 0.411i·13-s + 0.267·14-s + (0.265 − 0.873i)15-s + 0.250·16-s − 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403550901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403550901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2.13 + 0.651i)T \) |
| 7 | \( 1 + iT \) |
| 17 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.58iT - 3T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 + 4.11iT - 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + 8.92iT - 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 8.26iT - 47T^{2} \) |
| 53 | \( 1 - 4.09iT - 53T^{2} \) |
| 59 | \( 1 + 9.51T + 59T^{2} \) |
| 61 | \( 1 - 9.86T + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−9.527955687915654784707436466814, −9.316309028095978431779710522651, −8.172235281342471735073592716267, −7.51511549105425910758102837039, −6.74228991711210345760737747005, −5.57045580706686372250952659789, −4.62008955099790559163656054715, −4.09363417902726841328024424600, −3.19956390619743792427154581203, −0.908124392278879679092600143990,
0.952330962302198965540440821541, 2.06850922923517315346902717270, 3.32743027787136869246194298226, 4.09790129312164655676070301210, 5.24508420394466476768606258398, 6.46245063853352091345259217443, 7.19923467861710920100385644068, 7.905078207066046419110495596544, 8.781108548164108670878561884189, 9.594083990544641718135709675152
