| L(s) = 1 | − i·2-s + i·3-s − 4-s − 2.46i·5-s + 6-s + i·8-s − 9-s − 2.46·10-s + (3.20 + 0.834i)11-s − i·12-s + 1.32·13-s + 2.46·15-s + 16-s − 4.47·17-s + i·18-s − 4.43·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 1.10i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.778·10-s + (0.967 + 0.251i)11-s − 0.288i·12-s + 0.366·13-s + 0.635·15-s + 0.250·16-s − 1.08·17-s + 0.235i·18-s − 1.01·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01894130024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01894130024\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.20 - 0.834i)T \) |
| good | 5 | \( 1 + 2.46iT - 5T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 - 1.44iT - 29T^{2} \) |
| 31 | \( 1 + 2.71iT - 31T^{2} \) |
| 37 | \( 1 + 4.93T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 9.63iT - 43T^{2} \) |
| 47 | \( 1 + 0.767iT - 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 + 7.78iT - 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 1.72T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.92iT - 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 1.37iT - 97T^{2} \) |
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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
−8.864674893894993038015760427248, −8.633202538966542511718218362059, −7.68018209154148398490018347388, −6.41052157425625169156535173516, −5.83715940655592639650023690785, −4.56530246150245828234316498639, −4.43216320022940164063188336026, −3.57096913720453168920558032232, −2.27479792291193769249066538335, −1.38484152920714534173405932941,
0.00569917396992138716578137576, 1.68146445735302509241257848854, 2.68419300290381481393656873007, 3.80152183583438727910675262676, 4.40919976476626817966761836470, 5.87138823289534972134087012689, 6.17972976629415655886725933852, 6.93939467597035514628249762744, 7.35744681214273566439310124353, 8.480688131644906606172245580890
