L(s) = 1 | + i·2-s − 4-s + 5-s + (−1.80 − 1.93i)7-s − i·8-s + i·10-s − 3.87i·11-s + 1.60i·13-s + (1.93 − 1.80i)14-s + 16-s + 8.11·17-s − 2.63i·19-s − 20-s + 3.87·22-s − 5.47i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447·5-s + (−0.681 − 0.732i)7-s − 0.353i·8-s + 0.316i·10-s − 1.16i·11-s + 0.445i·13-s + (0.517 − 0.481i)14-s + 0.250·16-s + 1.96·17-s − 0.605i·19-s − 0.223·20-s + 0.825·22-s − 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36290 - 0.140813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36290 - 0.140813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.80 + 1.93i)T \) |
good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 8.11T + 17T^{2} \) |
| 19 | \( 1 + 2.63iT - 19T^{2} \) |
| 23 | \( 1 + 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 2.26iT - 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 + 11.8iT - 61T^{2} \) |
| 67 | \( 1 - 6.90T + 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.68iT - 97T^{2} \) |
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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
−10.28886445816755206566268530581, −9.740622520088579851833346522433, −8.728376652344661856174378291245, −7.899188728258031521779114112660, −6.88148817319108516824439086678, −6.15732528231902483755320301057, −5.28527100682008195137274494497, −4.00769381861254557940778529550, −2.97082609569663283758364471068, −0.832622115489086992223016384518,
1.51982949569219701646062882563, 2.79257968878886014722580671186, 3.75213266364597328300497716583, 5.24086526299986533811088304322, 5.80503984409341565909995805277, 7.15073092838133624684489809893, 8.099178849406798753093844283952, 9.242997332361519587613224068836, 9.886545191305335851375099642733, 10.31873420776268722651582106823