L(s) = 1 | + (2.05 + 1.18i)2-s + (1.82 + 3.15i)4-s + (−0.733 + 2.11i)5-s + (2.03 − 1.68i)7-s + 3.91i·8-s + (−4.01 + 3.47i)10-s + (−1.80 − 3.13i)11-s + 3.36i·13-s + (6.19 − 1.04i)14-s + (−1 + 1.73i)16-s + (−4.71 + 2.72i)17-s + (2.32 − 4.02i)19-s + (−8.00 + 1.53i)20-s − 8.58i·22-s + (−0.599 − 0.346i)23-s + ⋯ |
L(s) = 1 | + (1.45 + 0.840i)2-s + (0.911 + 1.57i)4-s + (−0.328 + 0.944i)5-s + (0.771 − 0.636i)7-s + 1.38i·8-s + (−1.27 + 1.09i)10-s + (−0.544 − 0.943i)11-s + 0.934i·13-s + (1.65 − 0.278i)14-s + (−0.250 + 0.433i)16-s + (−1.14 + 0.660i)17-s + (0.532 − 0.923i)19-s + (−1.79 + 0.343i)20-s − 1.83i·22-s + (−0.125 − 0.0722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00989 + 1.81076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00989 + 1.81076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.733 - 2.11i)T \) |
| 7 | \( 1 + (-2.03 + 1.68i)T \) |
good | 2 | \( 1 + (-2.05 - 1.18i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (1.80 + 3.13i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.36iT - 13T^{2} \) |
| 17 | \( 1 + (4.71 - 2.72i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.599 + 0.346i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.51 - 2.60i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 8.91iT - 43T^{2} \) |
| 47 | \( 1 + (-3.38 - 1.95i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.57 - 3.21i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.388 + 0.673i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 - 4.88i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.9 - 6.90i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 + (1.31 - 0.760i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.96 - 8.60i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.76iT - 83T^{2} \) |
| 89 | \( 1 + (-1.80 + 3.13i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−11.84473739763587631471472039753, −11.27301919481218574818048694570, −10.40627326755499808003002707466, −8.642633732977568400288144269313, −7.61381213786208601567572848803, −6.85241270800232099098131688789, −6.03643018006721322503543158232, −4.72805538357340757573578891290, −3.94769618549323138601625077847, −2.68812981909094111058937318618,
1.70525050756450954474049035199, 2.99699623307110268854300659435, 4.55428883083224190097975546830, 4.94990171297975299176895762363, 5.95358054828098573469650250414, 7.62326624690992000398556420769, 8.610699559943656810356025125772, 9.886167179214252049582411524761, 10.89879290145021162910352623727, 11.81653390239666816705638557835