| L(s) = 1 | + (−0.766 + 0.492i)2-s + (0.142 − 0.989i)3-s + (−0.486 + 1.06i)4-s + (0.959 − 0.281i)5-s + (0.378 + 0.828i)6-s + (−1.59 − 1.83i)7-s + (−0.410 − 2.85i)8-s + (−0.959 − 0.281i)9-s + (−0.596 + 0.688i)10-s + (3.16 + 2.03i)11-s + (0.984 + 0.632i)12-s + (3.24 − 3.74i)13-s + (2.12 + 0.623i)14-s + (−0.142 − 0.989i)15-s + (0.189 + 0.218i)16-s + (−1.16 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.348i)2-s + (0.0821 − 0.571i)3-s + (−0.243 + 0.532i)4-s + (0.429 − 0.125i)5-s + (0.154 + 0.338i)6-s + (−0.601 − 0.694i)7-s + (−0.145 − 1.01i)8-s + (−0.319 − 0.0939i)9-s + (−0.188 + 0.217i)10-s + (0.954 + 0.613i)11-s + (0.284 + 0.182i)12-s + (0.898 − 1.03i)13-s + (0.567 + 0.166i)14-s + (−0.0367 − 0.255i)15-s + (0.0474 + 0.0547i)16-s + (−0.281 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.935571 - 0.303030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935571 - 0.303030i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.99 - 2.65i)T \) |
| good | 2 | \( 1 + (0.766 - 0.492i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (1.59 + 1.83i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.16 - 2.03i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.24 + 3.74i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.16 + 2.54i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.28 + 2.81i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 2.75i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.07 + 7.48i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.0620 - 0.0182i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.54 + 1.62i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.04 + 7.23i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 + (-2.07 - 2.39i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (7.60 - 8.78i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.901 - 6.27i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (10.6 - 6.81i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (10.1 - 6.55i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.553 - 1.21i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.81 - 5.56i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.96 - 0.575i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.13 + 7.90i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.321 - 0.0943i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.47245520908404553145013180414, −10.26675594462014456205027259189, −9.305249053111350714724506383814, −8.726274383685504509303501889047, −7.41616920171369495697487224948, −6.99826988237387503359612294083, −5.84897083089885367494224680684, −4.20084559093025027195745292469, −3.03236790393562056775373101030, −0.925451863908976900645750439643,
1.58313548549735354799848251448, 3.17760794906643614440673015032, 4.55962848837056742978171222869, 5.93882304870666043663789146471, 6.42098763705283371730386320477, 8.412509332250749236418869173372, 9.099840157324006947617362990211, 9.551036964020546401489562427652, 10.66193920397183988463025380382, 11.25126587468540058651777213007
