| L(s) = 1 | − i·3-s + 2·4-s − 4i·7-s + 2·9-s − 6·11-s − 2i·12-s − i·13-s + 4·16-s + 6i·17-s + 4·19-s − 4·21-s − 3i·23-s − 5i·27-s − 8i·28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 4-s − 1.51i·7-s + 0.666·9-s − 1.80·11-s − 0.577i·12-s − 0.277i·13-s + 16-s + 1.45i·17-s + 0.917·19-s − 0.872·21-s − 0.625i·23-s − 0.962i·27-s − 1.51i·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35090 - 0.834907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35090 - 0.834907i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−11.28056960963013248711479131282, −10.38532992459382960078819564760, −10.17886255147936818805253913674, −8.083720860975592525034512941519, −7.57271774595739394408636961578, −6.84822816718476129432659610580, −5.70478599014574589119459369342, −4.21315269890944364716835223631, −2.78273445202813028289465280478, −1.26623473151111815808494300679,
2.22500974514134351030333953752, 3.15058073148390548229671054324, 5.07024791076197818923230671886, 5.61468073661378941204387736572, 7.07307005488618920370399863771, 7.85186770058065637042496647655, 9.155262343305100798286275974271, 9.942892024981197959889933819692, 10.87570316973455003894197143237, 11.72804645703094848947143972813
