| L(s) = 1 | + 5-s − 3·9-s + 11-s − 8·19-s + 8·23-s + 25-s + 10·29-s + 8·31-s − 10·37-s + 2·41-s + 6·43-s − 3·45-s − 8·47-s + 14·53-s + 55-s − 4·59-s − 10·61-s − 4·67-s + 8·73-s + 4·79-s + 9·81-s + 10·83-s − 6·89-s − 8·95-s + 10·97-s − 3·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.301·11-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s − 1.64·37-s + 0.312·41-s + 0.914·43-s − 0.447·45-s − 1.16·47-s + 1.92·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.936·73-s + 0.450·79-s + 81-s + 1.09·83-s − 0.635·89-s − 0.820·95-s + 1.01·97-s − 0.301·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.185708845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185708845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.75578511904879, −14.07709134623210, −13.75858801415513, −13.23661237953204, −12.56439611261555, −12.12201156416422, −11.66888808931491, −10.84737416955801, −10.61667001510512, −10.13728755443798, −9.174240882848992, −8.981375017298252, −8.349083408067893, −8.007913736754194, −6.917352205153887, −6.623789989193544, −6.137253545577185, −5.424872040190358, −4.812831782352319, −4.338950476963628, −3.423445324323200, −2.772183775814775, −2.330045585915033, −1.360226839834188, −0.5496556987555217,
0.5496556987555217, 1.360226839834188, 2.330045585915033, 2.772183775814775, 3.423445324323200, 4.338950476963628, 4.812831782352319, 5.424872040190358, 6.137253545577185, 6.623789989193544, 6.917352205153887, 8.007913736754194, 8.349083408067893, 8.981375017298252, 9.174240882848992, 10.13728755443798, 10.61667001510512, 10.84737416955801, 11.66888808931491, 12.12201156416422, 12.56439611261555, 13.23661237953204, 13.75858801415513, 14.07709134623210, 14.75578511904879
