| L(s) = 1 | + (1.57 − 0.714i)3-s + 1.85i·5-s − 0.707i·7-s + (1.97 − 2.25i)9-s + 4.63·11-s − 1.78·13-s + (1.32 + 2.91i)15-s + 1.47i·17-s − i·19-s + (−0.505 − 1.11i)21-s + 2.68·23-s + 1.57·25-s + (1.51 − 4.97i)27-s − 2.44i·29-s + 6.63i·31-s + ⋯ |
| L(s) = 1 | + (0.910 − 0.412i)3-s + 0.827i·5-s − 0.267i·7-s + (0.659 − 0.751i)9-s + 1.39·11-s − 0.494·13-s + (0.341 + 0.753i)15-s + 0.358i·17-s − 0.229i·19-s + (−0.110 − 0.243i)21-s + 0.559·23-s + 0.315·25-s + (0.290 − 0.956i)27-s − 0.453i·29-s + 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33241 - 0.114757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33241 - 0.114757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 + 0.714i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.85iT - 5T^{2} \) |
| 7 | \( 1 + 0.707iT - 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 1.47iT - 17T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 6.63iT - 31T^{2} \) |
| 37 | \( 1 + 3.04T + 37T^{2} \) |
| 41 | \( 1 + 4.95iT - 41T^{2} \) |
| 43 | \( 1 - 6.80iT - 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 3.89iT - 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.84iT - 67T^{2} \) |
| 71 | \( 1 + 6.09T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 2.21iT - 79T^{2} \) |
| 83 | \( 1 + 2.65T + 83T^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.45T + 97T^{2} \) |
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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
−10.02550908094659670642526090596, −9.104249741532741410769490312146, −8.538417094705790454617319883324, −7.24156965896627574843911491304, −7.01041324916443318219915999821, −6.03566464955639430254257594661, −4.47224131889973263913072775705, −3.54820999543284954473126862131, −2.65049809608701253288821811111, −1.34993559824570244569950862500,
1.33915255479307901083481606092, 2.64310838592331052078070438260, 3.85825277539080113339980769062, 4.60467518493860939527486886001, 5.58096042266268999547834662907, 6.86118258596362858914279277744, 7.69259462049751344020790150942, 8.827124290994063839447986870037, 9.024945786022058696718034896380, 9.834062854587772740134662708184
