| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 13-s + 16-s − 17-s + 18-s + 24-s − 26-s + 27-s − 2·29-s + 8·31-s + 32-s − 34-s + 36-s − 10·37-s − 39-s − 6·41-s + 4·43-s + 8·47-s + 48-s − 7·49-s − 51-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 + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s − 1.64·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.144·48-s − 49-s − 0.140·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.20329725113418, −14.71959561005305, −14.17798779724440, −13.61609269080209, −13.43538222467676, −12.64224281316820, −12.17899456212902, −11.78536722386529, −11.05130127583260, −10.47967054114574, −10.02020913229580, −9.391707531940025, −8.650955666220080, −8.361129934921532, −7.415662434407779, −7.209326761460536, −6.435111857163669, −5.855191740790737, −5.193231834977718, −4.469793902482641, −4.121371600684159, −3.129240180385500, −2.886172623897793, −1.943999798815563, −1.329109863501309, 0,
1.329109863501309, 1.943999798815563, 2.886172623897793, 3.129240180385500, 4.121371600684159, 4.469793902482641, 5.193231834977718, 5.855191740790737, 6.435111857163669, 7.209326761460536, 7.415662434407779, 8.361129934921532, 8.650955666220080, 9.391707531940025, 10.02020913229580, 10.47967054114574, 11.05130127583260, 11.78536722386529, 12.17899456212902, 12.64224281316820, 13.43538222467676, 13.61609269080209, 14.17798779724440, 14.71959561005305, 15.20329725113418
