| L(s) = 1 | + (7.79 − 4.5i)5-s + (−2.5 + 4.33i)7-s + (−101. − 58.5i)11-s + (17 + 29.4i)13-s + 450i·17-s − 64·19-s + (530. − 306i)23-s + (−272 + 471. i)25-s + (−919. − 531i)29-s + (348.5 + 603. i)31-s + 45.0i·35-s − 748·37-s + (592. − 342i)41-s + (−1.30e3 + 2.26e3i)43-s + (2.29e3 + 1.32e3i)47-s + ⋯ |
| L(s) = 1 | + (0.311 − 0.179i)5-s + (−0.0510 + 0.0883i)7-s + (−0.837 − 0.483i)11-s + (0.100 + 0.174i)13-s + 1.55i·17-s − 0.177·19-s + (1.00 − 0.578i)23-s + (−0.435 + 0.753i)25-s + (−1.09 − 0.631i)29-s + (0.362 + 0.628i)31-s + 0.0367i·35-s − 0.546·37-s + (0.352 − 0.203i)41-s + (−0.707 + 1.22i)43-s + (1.03 + 0.598i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.100886732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100886732\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.79 + 4.5i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (2.5 - 4.33i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (101. + 58.5i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-17 - 29.4i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 450iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 64T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-530. + 306i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (919. + 531i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-348.5 - 603. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 748T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-592. + 342i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.30e3 - 2.26e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-2.29e3 - 1.32e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.07e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (5.03e3 - 2.90e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.20e3 - 5.54e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.60e3 - 4.51e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 6.57e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.51e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (3.75e3 - 6.49e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-4.74e3 - 2.74e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (5.28e3 - 9.15e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−11.04180171563725793623889847633, −10.49136850057957211483638509500, −9.327113079496955135831507981125, −8.483250715616078227439003459923, −7.53023605800599133112164574034, −6.24291170867259545871993361866, −5.44240297028065295654696892217, −4.15293334586567211237677700509, −2.82110133782916402716509672270, −1.41225581314495066999077172302,
0.32439862576861860275043806359, 2.08098374861850014837142379707, 3.24945992923008744686857057921, 4.76114503288444828360122481648, 5.63793315064998446251775143456, 6.95844107678543247448711212068, 7.66823073958378999827340186035, 8.937384902908502837959565139351, 9.794873088592668214886150404632, 10.64322008180985428352173707974
