| L(s) = 1 | + (1.83 − 1.17i)2-s + (0.142 − 0.989i)3-s + (1.13 − 2.48i)4-s + (0.959 − 0.281i)5-s + (−0.903 − 1.97i)6-s + (−0.659 − 0.761i)7-s + (−0.227 − 1.57i)8-s + (−0.959 − 0.281i)9-s + (1.42 − 1.64i)10-s + (−0.301 − 0.193i)11-s + (−2.29 − 1.47i)12-s + (−0.275 + 0.318i)13-s + (−2.10 − 0.617i)14-s + (−0.142 − 0.989i)15-s + (1.30 + 1.50i)16-s + (0.200 + 0.439i)17-s + ⋯ |
| L(s) = 1 | + (1.29 − 0.831i)2-s + (0.0821 − 0.571i)3-s + (0.567 − 1.24i)4-s + (0.429 − 0.125i)5-s + (−0.368 − 0.807i)6-s + (−0.249 − 0.287i)7-s + (−0.0802 − 0.558i)8-s + (−0.319 − 0.0939i)9-s + (0.450 − 0.519i)10-s + (−0.0907 − 0.0583i)11-s + (−0.663 − 0.426i)12-s + (−0.0765 + 0.0882i)13-s + (−0.561 − 0.164i)14-s + (−0.0367 − 0.255i)15-s + (0.326 + 0.376i)16-s + (0.0487 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75250 - 2.01348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75250 - 2.01348i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.468 - 4.77i)T \) |
| good | 2 | \( 1 + (-1.83 + 1.17i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (0.659 + 0.761i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.301 + 0.193i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.275 - 0.318i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.200 - 0.439i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.0275 - 0.0603i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.27 + 4.98i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.00 - 6.97i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.48 + 1.31i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.48 + 1.02i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.821 - 5.71i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.09T + 47T^{2} \) |
| 53 | \( 1 + (2.16 + 2.50i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.19 + 3.69i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.27 - 8.89i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (11.9 - 7.66i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.03 + 3.23i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.752 + 1.64i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (1.10 - 1.27i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (4.64 + 1.36i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.17 + 8.14i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (13.1 - 3.85i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.56579704019689162366103698905, −10.63924031087121502538622111714, −9.705672561982127134565630077857, −8.451345283610810053300652195800, −7.20901958318857638272986137101, −6.07087362535435226625816148564, −5.22853656366766995286383398320, −3.98094481999956086592242035264, −2.86405808505635252492509639865, −1.61138581546755767877685890129,
2.68817737969874265026759769559, 3.87811358610917456253676740699, 4.92555679205873026348138873164, 5.77848501744884709520750503468, 6.62455315898188384419948221991, 7.70517411203983597172621628742, 8.944794531407689606904962285266, 9.918969724445228160721673041847, 10.89064796388784786996563136279, 12.12028706065296143151708734324
