| L(s) = 1 | + (−1.08 − 0.761i)2-s + (0.660 − 1.60i)3-s + (−0.0804 − 0.221i)4-s + (1.23 − 1.86i)5-s + (−1.93 + 1.23i)6-s + (0.827 + 1.77i)7-s + (−0.768 + 2.86i)8-s + (−2.12 − 2.11i)9-s + (−2.76 + 1.08i)10-s + (−2.76 − 3.29i)11-s + (−0.407 − 0.0172i)12-s + (2.90 + 4.15i)13-s + (0.451 − 2.56i)14-s + (−2.16 − 3.20i)15-s + (2.66 − 2.23i)16-s + (−0.828 − 3.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.769 − 0.538i)2-s + (0.381 − 0.924i)3-s + (−0.0402 − 0.110i)4-s + (0.551 − 0.833i)5-s + (−0.791 + 0.505i)6-s + (0.312 + 0.670i)7-s + (−0.271 + 1.01i)8-s + (−0.708 − 0.705i)9-s + (−0.873 + 0.344i)10-s + (−0.833 − 0.993i)11-s + (−0.117 − 0.00499i)12-s + (0.806 + 1.15i)13-s + (0.120 − 0.684i)14-s + (−0.560 − 0.828i)15-s + (0.665 − 0.558i)16-s + (−0.200 − 0.749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.385635 - 0.752733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.385635 - 0.752733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.660 + 1.60i)T \) |
| 5 | \( 1 + (-1.23 + 1.86i)T \) |
| good | 2 | \( 1 + (1.08 + 0.761i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.827 - 1.77i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (2.76 + 3.29i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.90 - 4.15i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.828 + 3.09i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.07 - 2.36i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 6.50i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 0.828i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-6.96 + 1.86i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-8.33 - 1.47i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.112 + 1.28i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (7.46 - 3.48i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.947 + 0.947i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.72 + 3.12i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.90 + 2.87i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (7.90 - 5.53i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-4.92 - 2.84i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.93 - 1.32i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.410 - 0.0724i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.31 - 7.59i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (0.974 + 1.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.4 - 1.17i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.91367658790391417727404483829, −11.69530641071313294473419824385, −10.98176171028967248877855206889, −9.308526850978712510502182704852, −8.892903618868934054203350300590, −8.009186588616634728327075304729, −6.22857402364380301696442453055, −5.19238674874151718463423536290, −2.59693560811058655491429749554, −1.23660005950133431018710286338,
2.94556054987299225144892204031, 4.39755040275831450605930647962, 6.09038107075962726645398179901, 7.50328055711080531059022987259, 8.240805960190396419976996160481, 9.505566091230385214389133607507, 10.35403744770705492220687688504, 10.90098527445097283757890217341, 12.87280355844606894534854445156, 13.64926104024473017328998367456
