| L(s) = 1 | + 5-s − 11-s − 6·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 10·37-s + 10·41-s − 4·47-s + 10·53-s − 55-s + 4·59-s + 2·61-s − 6·65-s + 8·67-s + 14·73-s + 16·79-s + 8·83-s − 6·85-s − 6·89-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 1.64·37-s + 1.56·41-s − 0.583·47-s + 1.37·53-s − 0.134·55-s + 0.520·59-s + 0.256·61-s − 0.744·65-s + 0.977·67-s + 1.63·73-s + 1.80·79-s + 0.878·83-s − 0.650·85-s − 0.635·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.967224724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967224724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.39986524660625, −12.14095199689604, −11.53166655246943, −11.10850697894670, −10.65662671136735, −10.09403564073144, −9.824296857853796, −9.293771786180090, −8.957607210574143, −8.349131930510628, −7.930623423145795, −7.313190258170160, −6.885107590711928, −6.659460789546637, −5.963361353912928, −5.304022488060702, −5.023579870220468, −4.624619070216001, −4.036709615611832, −3.280383316661625, −2.796839809081785, −2.214228977641239, −2.008333676341534, −0.8869801645044477, −0.5332272710462962,
0.5332272710462962, 0.8869801645044477, 2.008333676341534, 2.214228977641239, 2.796839809081785, 3.280383316661625, 4.036709615611832, 4.624619070216001, 5.023579870220468, 5.304022488060702, 5.963361353912928, 6.659460789546637, 6.885107590711928, 7.313190258170160, 7.930623423145795, 8.349131930510628, 8.957607210574143, 9.293771786180090, 9.824296857853796, 10.09403564073144, 10.65662671136735, 11.10850697894670, 11.53166655246943, 12.14095199689604, 12.39986524660625
