L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s − 2·13-s + 14-s + 16-s − 2·17-s + 20-s + 22-s − 6·23-s + 25-s + 2·26-s − 28-s + 8·29-s − 8·31-s − 32-s + 2·34-s − 35-s − 2·37-s − 40-s − 2·41-s − 44-s + 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s − 0.158·40-s − 0.312·41-s − 0.150·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.39553541576281, −12.98375839520147, −12.49830735285967, −12.00830729084904, −11.58738938184564, −10.97292406558311, −10.52904545321211, −9.971215755518967, −9.837511432976847, −9.172953798206489, −8.743909286343057, −8.211247172460259, −7.759768654504119, −7.180089716159998, −6.699236532539224, −6.209815955667606, −5.801839806811951, −4.992377858864109, −4.738864677161208, −3.796979649739252, −3.341073663626162, −2.618689848860313, −2.097535854727071, −1.628760792250765, −0.6699872148104399, 0,
0.6699872148104399, 1.628760792250765, 2.097535854727071, 2.618689848860313, 3.341073663626162, 3.796979649739252, 4.738864677161208, 4.992377858864109, 5.801839806811951, 6.209815955667606, 6.699236532539224, 7.180089716159998, 7.759768654504119, 8.211247172460259, 8.743909286343057, 9.172953798206489, 9.837511432976847, 9.971215755518967, 10.52904545321211, 10.97292406558311, 11.58738938184564, 12.00830729084904, 12.49830735285967, 12.98375839520147, 13.39553541576281