| L(s) = 1 | − 2.44i·3-s + (−1.73 − 1.41i)5-s − 1.41i·7-s − 2.99·9-s − 2·11-s − 5.65i·13-s + (−3.46 + 4.24i)15-s − 4.89i·17-s + 6·19-s − 3.46·21-s + 7.07i·23-s + (0.999 + 4.89i)25-s − 6.92·29-s − 6.92·31-s + 4.89i·33-s + ⋯ |
| L(s) = 1 | − 1.41i·3-s + (−0.774 − 0.632i)5-s − 0.534i·7-s − 0.999·9-s − 0.603·11-s − 1.56i·13-s + (−0.894 + 1.09i)15-s − 1.18i·17-s + 1.37·19-s − 0.755·21-s + 1.47i·23-s + (0.199 + 0.979i)25-s − 1.28·29-s − 1.24·31-s + 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9029993104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9029993104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 4.24iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 2.44iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−9.058077593229949441266667822961, −7.900600055786704558960587165192, −7.50284571514829397243496215635, −7.24597450048125207493348947365, −5.63887906470805991706518211498, −5.25548343798241346493127484307, −3.75947497632796213646139191821, −2.82178546428874127849353441339, −1.31469913140239426573327330591, −0.40344577130341762491147327837,
2.21617798146916164013296282594, 3.45057419874259817876337104033, 4.05190285209349208127745644965, 4.91571240029925803441555444169, 5.86035809748366187555233794360, 6.90727219689775014162525202158, 7.79175261363392901294073473707, 8.819241650787510371577021426496, 9.301595801333038797001086201135, 10.29182014416908893484086404428
