| L(s) = 1 | + (1.99 − 0.0807i)2-s + (2.72 + 1.26i)3-s + (3.98 − 0.322i)4-s + (1.26 + 7.17i)5-s + (5.54 + 2.30i)6-s + (−9.30 − 7.80i)7-s + (7.94 − 0.967i)8-s + (5.81 + 6.86i)9-s + (3.10 + 14.2i)10-s + (−2.17 + 12.3i)11-s + (11.2 + 4.14i)12-s + (7.39 − 20.3i)13-s + (−19.2 − 14.8i)14-s + (−5.59 + 21.1i)15-s + (15.7 − 2.57i)16-s + (2.11 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0403i)2-s + (0.907 + 0.420i)3-s + (0.996 − 0.0806i)4-s + (0.252 + 1.43i)5-s + (0.923 + 0.383i)6-s + (−1.32 − 1.11i)7-s + (0.992 − 0.120i)8-s + (0.646 + 0.762i)9-s + (0.310 + 1.42i)10-s + (−0.198 + 1.12i)11-s + (0.938 + 0.345i)12-s + (0.569 − 1.56i)13-s + (−1.37 − 1.06i)14-s + (−0.373 + 1.40i)15-s + (0.986 − 0.160i)16-s + (0.124 − 0.0719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.29274 + 1.16746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.29274 + 1.16746i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 + 0.0807i)T \) |
| 3 | \( 1 + (-2.72 - 1.26i)T \) |
| good | 5 | \( 1 + (-1.26 - 7.17i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (9.30 + 7.80i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.17 - 12.3i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-7.39 + 20.3i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 1.22i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (16.9 + 9.81i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (10.3 + 12.3i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-0.310 + 0.113i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (0.992 - 0.832i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-52.6 + 30.3i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-6.40 + 17.5i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (7.14 + 1.26i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (9.16 - 10.9i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 81.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-0.313 - 1.77i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (28.7 - 34.2i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-5.96 + 16.3i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (70.5 - 40.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-16.4 + 28.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (102. - 37.1i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (5.62 - 2.04i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-19.3 - 11.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (6.97 - 39.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.78683309799794922071203699914, −10.82007603968095002438850902731, −10.42919249377678647691993644617, −9.746805128009550084754028533702, −7.76147251021677823846883741160, −7.03564494789107207345263167455, −6.12240517052827160307272072759, −4.32196830477364676485809774634, −3.32342851076949747725595239970, −2.55502643799242344690759861171,
1.72950290585567927004722454151, 3.13398179266473377475890530674, 4.32013611264487677941193143296, 5.89646095180456638581724454136, 6.44038631126702099219777218939, 8.159124952349468589045196881550, 8.920868092067373865865676108597, 9.720708376756127085614270830793, 11.55097653308947778212261838903, 12.37680816679015561593397417371
