L(s) = 1 | + (−1.41 − 0.817i)2-s + (−1.73 − 0.0392i)3-s + (0.335 + 0.581i)4-s + (0.5 + 0.866i)5-s + (2.41 + 1.47i)6-s + (−2.43 − 1.03i)7-s + 2.17i·8-s + (2.99 + 0.135i)9-s − 1.63i·10-s + (1.25 + 0.727i)11-s + (−0.558 − 1.02i)12-s + (−4.17 + 2.40i)13-s + (2.59 + 3.45i)14-s + (−0.831 − 1.51i)15-s + (2.44 − 4.23i)16-s + 6.37·17-s + ⋯ |
L(s) = 1 | + (−1.00 − 0.577i)2-s + (−0.999 − 0.0226i)3-s + (0.167 + 0.290i)4-s + (0.223 + 0.387i)5-s + (0.987 + 0.600i)6-s + (−0.920 − 0.391i)7-s + 0.767i·8-s + (0.998 + 0.0453i)9-s − 0.516i·10-s + (0.379 + 0.219i)11-s + (−0.161 − 0.294i)12-s + (−1.15 + 0.667i)13-s + (0.694 + 0.923i)14-s + (−0.214 − 0.392i)15-s + (0.611 − 1.05i)16-s + 1.54·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.471312 - 0.151523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.471312 - 0.151523i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.73 + 0.0392i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.43 + 1.03i)T \) |
good | 2 | \( 1 + (1.41 + 0.817i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.25 - 0.727i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.17 - 2.40i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 + 1.12iT - 19T^{2} \) |
| 23 | \( 1 + (-7.62 + 4.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.63 + 2.09i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.94 + 4.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 + (-4.21 - 7.30i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.739 - 1.28i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.46iT - 53T^{2} \) |
| 59 | \( 1 + (1.63 + 2.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.44 - 3.72i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.282 + 0.489i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.7iT - 71T^{2} \) |
| 73 | \( 1 + 6.61iT - 73T^{2} \) |
| 79 | \( 1 + (3.80 - 6.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.62 + 2.82i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 + (-10.0 - 5.79i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.38978406658701234405948441832, −10.46769090064371983338439245299, −9.802704788142026762965455837462, −9.316721730275194313737986064993, −7.65180231168056999450918143504, −6.82782827358492822649781984110, −5.74212520195348291750259208725, −4.48902066562425897091380714812, −2.68007210697891825965522877393, −0.897871452008684835929230705416,
0.868674160002360936869013418212, 3.40106334369086414307379846782, 5.10829597678647380736745020939, 5.99583895306399501062022877055, 7.02608558176879316909355229038, 7.80901365048513042788298480632, 9.158966837121289202875072390732, 9.738131144973666275749883559761, 10.44162224603531258855868211583, 11.83832130653808253646535023283