L(s) = 1 | + 5-s + 4·11-s + 6·13-s + 2·17-s + 4·23-s + 25-s − 2·29-s − 8·31-s + 2·37-s − 10·41-s − 43-s − 4·47-s − 7·49-s + 14·53-s + 4·55-s − 4·59-s − 2·61-s + 6·65-s + 4·67-s + 14·73-s − 8·79-s + 8·83-s + 2·85-s + 10·89-s − 6·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 1.56·41-s − 0.152·43-s − 0.583·47-s − 49-s + 1.92·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s + 0.744·65-s + 0.488·67-s + 1.63·73-s − 0.900·79-s + 0.878·83-s + 0.216·85-s + 1.05·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.411866808\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.411866808\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.12455967801460, −14.51953928876252, −14.03973596949295, −13.52711311094343, −13.02972393157024, −12.57159814726218, −11.77425756694658, −11.36766652542293, −10.91381921873486, −10.26587873091576, −9.673047380324601, −9.018935374311200, −8.776782360793412, −8.101948399418937, −7.340274977770937, −6.671653987035533, −6.324089364196231, −5.623252691510368, −5.115207224288102, −4.236897877652614, −3.518090500283848, −3.286181369123609, −2.021367196858924, −1.487000800013630, −0.7559404570426278,
0.7559404570426278, 1.487000800013630, 2.021367196858924, 3.286181369123609, 3.518090500283848, 4.236897877652614, 5.115207224288102, 5.623252691510368, 6.324089364196231, 6.671653987035533, 7.340274977770937, 8.101948399418937, 8.776782360793412, 9.018935374311200, 9.673047380324601, 10.26587873091576, 10.91381921873486, 11.36766652542293, 11.77425756694658, 12.57159814726218, 13.02972393157024, 13.52711311094343, 14.03973596949295, 14.51953928876252, 15.12455967801460