L(s) = 1 | − 3i·3-s − i·7-s − 6·9-s − 5·11-s − 5i·13-s + 7i·17-s + 2·19-s − 3·21-s − 2i·23-s + 9i·27-s − 7·29-s + 4·31-s + 15i·33-s + 6i·37-s − 15·39-s + ⋯ |
L(s) = 1 | − 1.73i·3-s − 0.377i·7-s − 2·9-s − 1.50·11-s − 1.38i·13-s + 1.69i·17-s + 0.458·19-s − 0.654·21-s − 0.417i·23-s + 1.73i·27-s − 1.29·29-s + 0.718·31-s + 2.61i·33-s + 0.986i·37-s − 2.40·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4729179979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4729179979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13iT - 97T^{2} \) |
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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
−8.458500745950601494457365975183, −8.116839195717068430291532311998, −7.52092873245584717882288237667, −6.65987578942819927692899991328, −5.81848835328526736895234409555, −5.14392213520221523780011852313, −3.48323871218730750841209915753, −2.54634699089576332556645417472, −1.47810162955999494268069435192, −0.18358137541973819908023094126,
2.36356503599207999051583964960, 3.25862285975149404729416265583, 4.26048787228535012777427976783, 5.15342213409149017338338020428, 5.46012397543347808758332835704, 6.85775665142860311628507074144, 7.83595882214809795956506854805, 8.835040812046141637034691858472, 9.454077248532162809256220197409, 9.912945338922367994741340918009