| L(s) = 1 | + 5-s − 4·7-s + 6·13-s + 6·17-s + 4·19-s + 25-s − 2·29-s − 8·31-s − 4·35-s + 10·37-s + 6·41-s + 43-s + 8·47-s + 9·49-s + 6·53-s + 10·61-s + 6·65-s + 4·67-s + 8·71-s − 6·73-s + 16·83-s + 6·85-s − 6·89-s − 24·91-s + 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.66·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.937·41-s + 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.824·53-s + 1.28·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 1.75·83-s + 0.650·85-s − 0.635·89-s − 2.51·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-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
| \(L(1)\) |
\(\approx\) |
\(2.741377415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.741377415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.09424687147706, −14.51625510899355, −13.96886074441460, −13.42265714582410, −13.04398974193710, −12.57531814529759, −12.03310193557380, −11.22674065397251, −10.89392631993950, −10.03410189781656, −9.834786939968751, −9.099011744659736, −8.883519595697961, −7.824848896373332, −7.518749280077202, −6.653177586942116, −6.217528136129963, −5.599928343158904, −5.358615877114646, −3.946013993961104, −3.754689769422188, −3.049033996220869, −2.388013923113027, −1.254129428849334, −0.7171444965997016,
0.7171444965997016, 1.254129428849334, 2.388013923113027, 3.049033996220869, 3.754689769422188, 3.946013993961104, 5.358615877114646, 5.599928343158904, 6.217528136129963, 6.653177586942116, 7.518749280077202, 7.824848896373332, 8.883519595697961, 9.099011744659736, 9.834786939968751, 10.03410189781656, 10.89392631993950, 11.22674065397251, 12.03310193557380, 12.57531814529759, 13.04398974193710, 13.42265714582410, 13.96886074441460, 14.51625510899355, 15.09424687147706
