L(s) = 1 | − 5-s − 7-s − 6·11-s − 13-s − 4·17-s − 8·19-s − 8·23-s + 25-s + 2·29-s + 6·31-s + 35-s + 6·37-s + 6·43-s + 49-s + 4·53-s + 6·55-s + 8·59-s − 2·61-s + 65-s + 8·67-s − 10·73-s + 6·77-s + 8·79-s − 12·83-s + 4·85-s + 91-s + 8·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 0.970·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s + 0.169·35-s + 0.986·37-s + 0.914·43-s + 1/7·49-s + 0.549·53-s + 0.809·55-s + 1.04·59-s − 0.256·61-s + 0.124·65-s + 0.977·67-s − 1.17·73-s + 0.683·77-s + 0.900·79-s − 1.31·83-s + 0.433·85-s + 0.104·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.61850655346015, −13.79272557044657, −13.43811225204213, −12.91181183913702, −12.55332832328996, −12.04668485872113, −11.38545772439491, −10.84614111126090, −10.42183353006869, −10.03673628270558, −9.438293137323887, −8.651513008681332, −8.137030908426836, −8.032325762156504, −7.166336576835025, −6.699727052173845, −6.024458207921166, −5.641450153623430, −4.757089577673111, −4.335606184226692, −3.898630892518053, −2.843316628812874, −2.480054284407022, −1.963484033078940, −0.6096586401211770, 0,
0.6096586401211770, 1.963484033078940, 2.480054284407022, 2.843316628812874, 3.898630892518053, 4.335606184226692, 4.757089577673111, 5.641450153623430, 6.024458207921166, 6.699727052173845, 7.166336576835025, 8.032325762156504, 8.137030908426836, 8.651513008681332, 9.438293137323887, 10.03673628270558, 10.42183353006869, 10.84614111126090, 11.38545772439491, 12.04668485872113, 12.55332832328996, 12.91181183913702, 13.43811225204213, 13.79272557044657, 14.61850655346015