| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s − 6·13-s + 14-s + 16-s + 6·17-s + 3·18-s + 6·19-s − 20-s + 8·23-s + 25-s + 6·26-s − 28-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 35-s − 3·36-s − 4·37-s − 6·38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 1.37·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s − 1/2·36-s − 0.657·37-s − 0.973·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.47275411963440337794882943605, −7.16625091924132139551037508014, −6.06002593200817576417677308102, −5.36332490350365140438107205580, −4.89962138351509561722300789879, −3.42636580384166541105292756242, −3.17088884485128297302019586200, −2.23242126154818294190934771949, −0.992445695085263760308742122746, 0,
0.992445695085263760308742122746, 2.23242126154818294190934771949, 3.17088884485128297302019586200, 3.42636580384166541105292756242, 4.89962138351509561722300789879, 5.36332490350365140438107205580, 6.06002593200817576417677308102, 7.16625091924132139551037508014, 7.47275411963440337794882943605
