L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.28 − 1.83i)5-s + (0.0564 − 2.64i)7-s + (−0.707 − 0.707i)8-s + (−0.386 − 2.20i)10-s + 1.41·11-s + (−0.772 + 0.772i)13-s + (−1.83 − 1.91i)14-s − 1.00·16-s + (−0.546 − 0.546i)17-s − 5.95·19-s + (−1.83 − 1.28i)20-s + (1.00 − 1.00i)22-s + (5.23 + 5.23i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.574 − 0.818i)5-s + (0.0213 − 0.999i)7-s + (−0.250 − 0.250i)8-s + (−0.122 − 0.696i)10-s + 0.426·11-s + (−0.214 + 0.214i)13-s + (−0.489 − 0.510i)14-s − 0.250·16-s + (−0.132 − 0.132i)17-s − 1.36·19-s + (−0.409 − 0.287i)20-s + (0.213 − 0.213i)22-s + (1.09 + 1.09i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10853 - 1.67453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10853 - 1.67453i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.28 + 1.83i)T \) |
| 7 | \( 1 + (-0.0564 + 2.64i)T \) |
good | 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + (0.772 - 0.772i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.546 + 0.546i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 + (-5.23 - 5.23i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.992iT - 29T^{2} \) |
| 31 | \( 1 + 2.85iT - 31T^{2} \) |
| 37 | \( 1 + (-5.70 + 5.70i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.18iT - 41T^{2} \) |
| 43 | \( 1 + (2 + 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.32 - 7.32i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.40 - 2.40i)T + 53iT^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 3.63iT - 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.40iT - 79T^{2} \) |
| 83 | \( 1 + (8.79 - 8.79i)T - 83iT^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-7.76 - 7.76i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.42466306195013566236126328642, −9.507991150471484196063731664939, −8.870190873556390922664783789715, −7.61316047283797572949531307412, −6.59314167937302232887053537801, −5.61022499155591133516860998671, −4.55849351196707037529495890379, −3.86869290108987300676746215370, −2.27253446300269422527995873568, −0.981776416218539787518466293209,
2.19636878577497436883883748863, 3.11712275747352192265827389920, 4.50901899404890348036180904589, 5.55465885106946879436029634380, 6.41361047988139141180470824440, 6.96333743303747341006929846426, 8.312483403795791373958081174065, 8.954141397739901328172769872484, 10.02380759325507369182885395655, 10.91429839165861065256286603453