L(s) = 1 | + (−44.0 − 76.3i)5-s + (−14.5 + 25.1i)7-s + (−44.0 + 76.3i)11-s + (−164.5 − 284. i)13-s − 2.20e3·17-s + 1.79e3·19-s + (−1.80e3 − 3.13e3i)23-s + (−2.32e3 + 4.02e3i)25-s + (−705. + 1.22e3i)29-s + (−2.61e3 − 4.52e3i)31-s + 2.55e3·35-s + 8.78e3·37-s + (7.75e3 + 1.34e4i)41-s + (−9.98e3 + 1.72e4i)43-s + (5.42e3 − 9.39e3i)47-s + ⋯ |
L(s) = 1 | + (−0.788 − 1.36i)5-s + (−0.111 + 0.193i)7-s + (−0.109 + 0.190i)11-s + (−0.269 − 0.467i)13-s − 1.85·17-s + 1.14·19-s + (−0.712 − 1.23i)23-s + (−0.744 + 1.28i)25-s + (−0.155 + 0.269i)29-s + (−0.488 − 0.846i)31-s + 0.352·35-s + 1.05·37-s + (0.720 + 1.24i)41-s + (−0.823 + 1.42i)43-s + (0.358 − 0.620i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4744947847\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4744947847\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (44.0 + 76.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (14.5 - 25.1i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (44.0 - 76.3i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (164.5 + 284. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 2.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.80e3 + 3.13e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (705. - 1.22e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.61e3 + 4.52e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 8.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.75e3 - 1.34e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (9.98e3 - 1.72e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-5.42e3 + 9.39e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.94e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (2.86e3 + 4.96e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-534.5 + 925. i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.10e4 - 5.37e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.49e4 - 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.88e4 + 4.99e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 8.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.45e3 - 1.11e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−11.22407057484587034877800668733, −9.899894912493468821325368237508, −8.982762319870874236949850286753, −8.257537963660596713387573196138, −7.34669652043767713883976965375, −5.99093022276243437509658538085, −4.80054777104255867474469001166, −4.14530909256683027975470334511, −2.51855257169075631848876412270, −0.927100071519429410835367512197,
0.15194050770754977453500940621, 2.15074753552206696132617107140, 3.33989667236541766142447681075, 4.24242363037715205516459641607, 5.76582560718863177388792798310, 7.03069124297558231810007965495, 7.34382004979477732296104575439, 8.661506630627305407320923916818, 9.751984196054516759630984586214, 10.76773255741567601527259090928