L(s) = 1 | − 2.18i·5-s + (1 − 2.44i)7-s + 5.26i·11-s + 1.01i·13-s − 3.08i·17-s + 3.24·19-s + 0.903i·23-s + 0.242·25-s + 5.34·29-s + 5.24·31-s + (−5.34 − 2.18i)35-s + 37-s + 8.35i·41-s − 4.47i·43-s + 12.8·47-s + ⋯ |
L(s) = 1 | − 0.975i·5-s + (0.377 − 0.925i)7-s + 1.58i·11-s + 0.281i·13-s − 0.748i·17-s + 0.743·19-s + 0.188i·23-s + 0.0485·25-s + 0.992·29-s + 0.941·31-s + (−0.903 − 0.368i)35-s + 0.164·37-s + 1.30i·41-s − 0.682i·43-s + 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.012418122\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.012418122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 5 | \( 1 + 2.18iT - 5T^{2} \) |
| 11 | \( 1 - 5.26iT - 11T^{2} \) |
| 13 | \( 1 - 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 3.08iT - 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 - 0.903iT - 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 8.35iT - 41T^{2} \) |
| 43 | \( 1 + 4.47iT - 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.88iT - 67T^{2} \) |
| 71 | \( 1 - 2.18iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 9.37iT - 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.98iT - 89T^{2} \) |
| 97 | \( 1 + 6.33iT - 97T^{2} \) |
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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
−8.624356355037225521473805711270, −7.75772655225807916400513960961, −7.24961733922511828458188219036, −6.49782564338728490722398886399, −5.26070610997266333498647939629, −4.64374306709071557329941279827, −4.23500335880941127213608801185, −2.91175835356353368399335585846, −1.69354537167868441863769487856, −0.805454613890060925437154469751,
1.03064429535488964589658588265, 2.52390058660053924227946943497, 3.02104588870076831572568379640, 3.96982527596415849892598821957, 5.17793812250280011160131129726, 5.88506579742064536606287400488, 6.41062398081760844473146472813, 7.33820438372341476327529592924, 8.299308976561922338931248660943, 8.581978989689940955989537564330