| L(s) = 1 | + (4.45 − 3.48i)2-s + (7.72 − 31.0i)4-s + 40.4i·5-s + 148. i·7-s + (−73.7 − 165. i)8-s + (141. + 180. i)10-s + 608.·11-s + 883.·13-s + (517. + 661. i)14-s + (−904. − 479. i)16-s + 1.13e3i·17-s − 1.11e3i·19-s + (1.25e3 + 312. i)20-s + (2.71e3 − 2.11e3i)22-s − 1.34e3·23-s + ⋯ |
| L(s) = 1 | + (0.787 − 0.615i)2-s + (0.241 − 0.970i)4-s + 0.724i·5-s + 1.14i·7-s + (−0.407 − 0.913i)8-s + (0.446 + 0.570i)10-s + 1.51·11-s + 1.45·13-s + (0.705 + 0.902i)14-s + (−0.883 − 0.468i)16-s + 0.951i·17-s − 0.706i·19-s + (0.702 + 0.174i)20-s + (1.19 − 0.933i)22-s − 0.530·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.15955 - 0.386951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15955 - 0.386951i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.45 + 3.48i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 40.4iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 148. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 608.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 883.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.13e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.85e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 606. iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.12e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.24e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.62e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.34e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.50e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.35e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.57e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 6.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.45e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e5T + 8.58e9T^{2} \) |
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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
−12.65911545226025071065500913937, −11.59315601178496258570143447350, −11.02593137361109585024961861442, −9.652040175882788332083707737084, −8.601800438640013913750542779805, −6.52617192826649204047253926057, −5.97000992485310458369624072825, −4.20036304031634893693805742987, −2.98815528118352734369503820628, −1.49213774125702166260525825581,
1.15818913962509286636950076318, 3.63253587966488037086468834657, 4.45601626340765157938642649826, 5.99294454313270033859191899688, 7.02326014244676411406852327190, 8.257407092126953737869619561676, 9.319476480055264050077980826375, 10.99533415696947377857433577903, 11.95478295400473818894673823741, 13.03938572785576755603173759237
