| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 12-s − 13-s − 2·14-s + 16-s + 17-s + 18-s − 4·19-s + 2·21-s + 6·23-s − 24-s − 5·25-s − 26-s − 27-s − 2·28-s + 8·29-s + 2·31-s + 32-s + 34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 1.25·23-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.408078166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.408078166\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.20385918874798, −12.70994556559205, −12.43497654430157, −12.00675837845976, −11.37870923237264, −10.98999342858044, −10.55546405511706, −9.963943538204867, −9.601711227110101, −9.071863853022248, −8.291493007846760, −7.963204333821583, −7.139786590465800, −6.861860057221219, −6.339563666798102, −5.882651406406245, −5.423186351743692, −4.746354411050597, −4.362224824174179, −3.781450985830690, −3.173793347520337, −2.532257523625457, −2.105415553204664, −1.031778371179536, −0.5721175562456816,
0.5721175562456816, 1.031778371179536, 2.105415553204664, 2.532257523625457, 3.173793347520337, 3.781450985830690, 4.362224824174179, 4.746354411050597, 5.423186351743692, 5.882651406406245, 6.339563666798102, 6.861860057221219, 7.139786590465800, 7.963204333821583, 8.291493007846760, 9.071863853022248, 9.601711227110101, 9.963943538204867, 10.55546405511706, 10.98999342858044, 11.37870923237264, 12.00675837845976, 12.43497654430157, 12.70994556559205, 13.20385918874798
