| L(s) = 1 | + i·3-s − i·5-s − 4.59·7-s − 9-s + 4.88·11-s + 1.41·13-s + 15-s + 4.35i·17-s − 1.34·19-s − 4.59i·21-s + (0.751 − 4.73i)23-s − 25-s − i·27-s − 8.47·29-s − 4.83i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 1.73·7-s − 0.333·9-s + 1.47·11-s + 0.392·13-s + 0.258·15-s + 1.05i·17-s − 0.307·19-s − 1.00i·21-s + (0.156 − 0.987i)23-s − 0.200·25-s − 0.192i·27-s − 1.57·29-s − 0.867i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.234026764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234026764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-0.751 + 4.73i)T \) |
| good | 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.35iT - 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 4.83iT - 31T^{2} \) |
| 37 | \( 1 - 2.55iT - 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 43 | \( 1 - 9.76T + 43T^{2} \) |
| 47 | \( 1 + 5.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 + 1.36iT - 59T^{2} \) |
| 61 | \( 1 - 9.11iT - 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 - 0.721iT - 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 8.99iT - 89T^{2} \) |
| 97 | \( 1 + 1.26iT - 97T^{2} \) |
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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
−8.715030435250809571618220766030, −7.55500675248980354879959874448, −6.68664271781824472436905183478, −6.11845948463333135778278984924, −5.68808909983966616031818329696, −4.31868318525997678492052193038, −3.94117554133028077431189474050, −3.28019105444121065395255449503, −2.16226296604281673292580840737, −0.856929951495048280358264354636,
0.41666953472504857176761951352, 1.61573736948075945444917574408, 2.71860233946561594905677619301, 3.50135177724304672881922348396, 3.94808165558768381612154364727, 5.34923850897455873531946719688, 6.05608217197777027389068500282, 6.72470998203206727527367440193, 6.99965820589264122918540924612, 7.80207947432980368091232393178
