| L(s) = 1 | − 3i·3-s + 13i·7-s − 9·9-s − 6·11-s − 5i·13-s − 78i·17-s + 65·19-s + 39·21-s + 138i·23-s + 27i·27-s − 66·29-s − 299·31-s + 18i·33-s − 214i·37-s − 15·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.701i·7-s − 0.333·9-s − 0.164·11-s − 0.106i·13-s − 1.11i·17-s + 0.784·19-s + 0.405·21-s + 1.25i·23-s + 0.192i·27-s − 0.422·29-s − 1.73·31-s + 0.0949i·33-s − 0.950i·37-s − 0.0615·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.765812577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765812577\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13iT - 343T^{2} \) |
| 11 | \( 1 + 6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 78iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 65T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 66T + 2.43e4T^{2} \) |
| 31 | \( 1 + 299T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 360T + 6.89e4T^{2} \) |
| 43 | \( 1 - 203iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 78iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 636iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 786T + 2.05e5T^{2} \) |
| 61 | \( 1 - 467T + 2.26e5T^{2} \) |
| 67 | \( 1 - 217iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 360T + 3.57e5T^{2} \) |
| 73 | \( 1 - 286iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 272T + 4.93e5T^{2} \) |
| 83 | \( 1 - 498iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 511iT - 9.12e5T^{2} \) |
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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
−9.304671572168460896963637526057, −8.377873865423949269549078488977, −7.45189350362613265037587131558, −6.96858285063666269720380266050, −5.53752249821532651297309521044, −5.43561806443928604258198779634, −3.85531734801714879486104513827, −2.81146002385808647494552163618, −1.85351087563044109151742964443, −0.53245161503554094863085430488,
0.875339683900731278692729194884, 2.28346228497073484167026591380, 3.56313038592067178488492986196, 4.20517563942308964410112548510, 5.22855090208212762960310226340, 6.09925951464706752865615274493, 7.10233278537625003633354001684, 7.894970783161215208138701311093, 8.796939396151636279135447881435, 9.533044486428903219122951184794
