| L(s) = 1 | + 3-s − 5-s − 2.56·7-s + 9-s − 5.12·11-s + 2·13-s − 15-s + 4.56·17-s + 1.12·19-s − 2.56·21-s − 23-s + 25-s + 27-s + 8.56·29-s − 3.68·31-s − 5.12·33-s + 2.56·35-s + 0.561·37-s + 2·39-s + 3.43·41-s − 6.24·43-s − 45-s + 8·47-s − 0.438·49-s + 4.56·51-s + 0.561·53-s + 5.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.968·7-s + 0.333·9-s − 1.54·11-s + 0.554·13-s − 0.258·15-s + 1.10·17-s + 0.257·19-s − 0.558·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.58·29-s − 0.661·31-s − 0.891·33-s + 0.432·35-s + 0.0923·37-s + 0.320·39-s + 0.536·41-s − 0.952·43-s − 0.149·45-s + 1.16·47-s − 0.0626·49-s + 0.638·51-s + 0.0771·53-s + 0.690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648013076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648013076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 0.561T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 0.315T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.647628545503945209172627614231, −8.095151061967214097186832221014, −7.44256146520155822291031177380, −6.64127949462959056346337207786, −5.70736405581931435701035048599, −4.93677324504034779539789351783, −3.77912332907744467464802328522, −3.16867426027455014136865314918, −2.36597058229060676703122820748, −0.76050468548182086492798811168,
0.76050468548182086492798811168, 2.36597058229060676703122820748, 3.16867426027455014136865314918, 3.77912332907744467464802328522, 4.93677324504034779539789351783, 5.70736405581931435701035048599, 6.64127949462959056346337207786, 7.44256146520155822291031177380, 8.095151061967214097186832221014, 8.647628545503945209172627614231
