| L(s) = 1 | + (0.877 + 0.479i)2-s + (0.997 + 0.0713i)3-s + (0.540 + 0.841i)4-s + (−1.94 + 1.09i)5-s + (0.841 + 0.540i)6-s + (0.266 + 0.713i)7-s + (0.0713 + 0.997i)8-s + (0.989 + 0.142i)9-s + (−2.23 + 0.0273i)10-s + (−1.07 + 3.67i)11-s + (0.479 + 0.877i)12-s + (1.99 + 0.745i)13-s + (−0.108 + 0.753i)14-s + (−2.02 + 0.953i)15-s + (−0.415 + 0.909i)16-s + (−3.59 + 0.782i)17-s + ⋯ |
| L(s) = 1 | + (0.620 + 0.338i)2-s + (0.575 + 0.0411i)3-s + (0.270 + 0.420i)4-s + (−0.871 + 0.489i)5-s + (0.343 + 0.220i)6-s + (0.100 + 0.269i)7-s + (0.0252 + 0.352i)8-s + (0.329 + 0.0474i)9-s + (−0.707 + 0.00864i)10-s + (−0.325 + 1.10i)11-s + (0.138 + 0.253i)12-s + (0.554 + 0.206i)13-s + (−0.0289 + 0.201i)14-s + (−0.522 + 0.246i)15-s + (−0.103 + 0.227i)16-s + (−0.872 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33406 + 1.64825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33406 + 1.64825i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.877 - 0.479i)T \) |
| 3 | \( 1 + (-0.997 - 0.0713i)T \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| 23 | \( 1 + (-2.66 - 3.98i)T \) |
| good | 7 | \( 1 + (-0.266 - 0.713i)T + (-5.29 + 4.58i)T^{2} \) |
| 11 | \( 1 + (1.07 - 3.67i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.99 - 0.745i)T + (9.82 + 8.51i)T^{2} \) |
| 17 | \( 1 + (3.59 - 0.782i)T + (15.4 - 7.06i)T^{2} \) |
| 19 | \( 1 + (-0.402 + 0.258i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 2.02i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.58 - 1.83i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-2.32 - 3.10i)T + (-10.4 + 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.610 - 4.24i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.197 + 2.75i)T + (-42.5 - 6.11i)T^{2} \) |
| 47 | \( 1 + (3.20 + 3.20i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.42 + 0.530i)T + (40.0 - 34.7i)T^{2} \) |
| 59 | \( 1 + (-7.27 + 3.32i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.67 + 3.18i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.93 + 7.20i)T + (-36.2 - 56.3i)T^{2} \) |
| 71 | \( 1 + (-3.48 + 1.02i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.41 + 6.49i)T + (-66.4 - 30.3i)T^{2} \) |
| 79 | \( 1 + (-2.14 - 4.69i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.565 + 0.423i)T + (23.3 - 79.6i)T^{2} \) |
| 89 | \( 1 + (-0.972 - 1.12i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-8.22 - 6.15i)T + (27.3 + 93.0i)T^{2} \) |
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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
−10.88675079660208005915254463214, −9.826526901440630074802894974455, −8.771359725684546983751646030762, −7.978848487131323674226070788414, −7.18012104858944043039542794205, −6.48098235831284966390074730464, −5.07080340186644094800870445139, −4.20355053355823520308378477385, −3.28402394083036004744681809549, −2.12375823353648028622549282824,
0.890117829583994924491420643272, 2.64657418866376871345812751150, 3.66088722054803687888510764093, 4.44645408346355860227456201334, 5.52136431953088593032454336348, 6.68837412871517533988258870127, 7.69753029631374405673720821644, 8.544941441391966919930045003522, 9.163581891317609973902394402354, 10.56551339304187473792814041429
