| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 4·11-s − 12-s + 2·13-s + 16-s − 2·17-s + 18-s + 4·22-s − 4·23-s − 24-s + 2·26-s − 27-s − 6·29-s − 4·31-s + 32-s − 4·33-s − 2·34-s + 36-s − 6·37-s − 2·39-s − 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.64787676574418, −14.12362939293446, −13.66762718913267, −13.16473519116522, −12.65223282626241, −12.02848644497707, −11.74640969638816, −11.25219477281926, −10.69023806682180, −10.25337472570542, −9.524880178946046, −8.997435190470603, −8.493596128437553, −7.719181731783514, −7.087200321269083, −6.673247938281736, −6.155118385529278, −5.567961251422447, −5.136590950601129, −4.311182529711962, −3.778824740159889, −3.535373569915238, −2.364836167768192, −1.789972522486504, −1.083044673086203, 0,
1.083044673086203, 1.789972522486504, 2.364836167768192, 3.535373569915238, 3.778824740159889, 4.311182529711962, 5.136590950601129, 5.567961251422447, 6.155118385529278, 6.673247938281736, 7.087200321269083, 7.719181731783514, 8.493596128437553, 8.997435190470603, 9.524880178946046, 10.25337472570542, 10.69023806682180, 11.25219477281926, 11.74640969638816, 12.02848644497707, 12.65223282626241, 13.16473519116522, 13.66762718913267, 14.12362939293446, 14.64787676574418
