L(s) = 1 | + (0.0597 − 1.73i)3-s + (0.567 + 2.58i)7-s + (−2.99 − 0.206i)9-s + (−0.793 + 0.457i)11-s − 4.31i·13-s + (0.268 + 0.465i)17-s + (1.12 + 0.651i)19-s + (4.50 − 0.827i)21-s + (−6.91 − 3.99i)23-s + (−0.537 + 5.16i)27-s − 3.46i·29-s + (−5.56 + 3.21i)31-s + (0.745 + 1.40i)33-s + (2.52 − 4.36i)37-s + (−7.46 − 0.257i)39-s + ⋯ |
L(s) = 1 | + (0.0345 − 0.999i)3-s + (0.214 + 0.976i)7-s + (−0.997 − 0.0689i)9-s + (−0.239 + 0.138i)11-s − 1.19i·13-s + (0.0651 + 0.112i)17-s + (0.258 + 0.149i)19-s + (0.983 − 0.180i)21-s + (−1.44 − 0.832i)23-s + (−0.103 + 0.994i)27-s − 0.642i·29-s + (−0.998 + 0.576i)31-s + (0.129 + 0.243i)33-s + (0.414 − 0.718i)37-s + (−1.19 − 0.0412i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6345530712\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6345530712\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0597 + 1.73i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.567 - 2.58i)T \) |
good | 11 | \( 1 + (0.793 - 0.457i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.31iT - 13T^{2} \) |
| 17 | \( 1 + (-0.268 - 0.465i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 0.651i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.91 + 3.99i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.52 + 4.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 8.22T + 43T^{2} \) |
| 47 | \( 1 + (-2.17 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 0.984i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.15 + 12.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.38 + 4.84i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.05 + 10.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.943iT - 71T^{2} \) |
| 73 | \( 1 + (8.85 - 5.11i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.71 - 9.90i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + (-0.874 + 1.51i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
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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
−8.481086212007710301809424177330, −7.995184813092649063580564828635, −7.31573939121011223550620580979, −6.19567626056679730804895799359, −5.75731007220217196082878629366, −4.92162510989304197170608817320, −3.50753354513101344783164968780, −2.55713954803767385403075266859, −1.75843875672578954872063442092, −0.21005021570522893601256666549,
1.59602820781281079423993350700, 2.96093649580777173115534294772, 3.98036947427954122776446107741, 4.41061982768751309943905653106, 5.41891364959528820256189829131, 6.23320468405642290427832613235, 7.29715360428376586757795624101, 7.900977183514595293839147323215, 8.955524835916825128297420253142, 9.433403432681689275062917229756