| L(s) = 1 | − 9.30·2-s + 22.6·4-s − 174. i·5-s + 384.·8-s + 1.62e3i·10-s + 185.·11-s − 3.98e3i·13-s − 5.03e3·16-s + 7.04e3i·17-s − 4.69e3i·19-s − 3.95e3i·20-s − 1.72e3·22-s + 6.64e3·23-s − 1.49e4·25-s + 3.70e4i·26-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.353·4-s − 1.39i·5-s + 0.751·8-s + 1.62i·10-s + 0.139·11-s − 1.81i·13-s − 1.22·16-s + 1.43i·17-s − 0.684i·19-s − 0.494i·20-s − 0.162·22-s + 0.546·23-s − 0.954·25-s + 2.10i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4350161220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4350161220\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 9.30T + 64T^{2} \) |
| 5 | \( 1 + 174. iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 185.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.98e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.04e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.69e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 6.64e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.93e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.39e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.95e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 4.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.31e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.49e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.01e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.52e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.68e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 6.11e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.85e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 8.92e3iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 4.43e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 5.59e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.06e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.23e6iT - 8.32e11T^{2} \) |
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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
−9.450581150057753032570256147840, −8.507851482213740683971943669786, −8.260748180536790856001795216294, −7.17175845526052307657693445378, −5.67433913952259592863554857921, −4.88675639195391580438809838833, −3.67751393747086828393537507630, −1.91533606182803200528924018756, −0.871647903772042632494551239891, −0.18695980802146359532578176909,
1.37891429255046378019131537851, 2.45409835118049573935367522791, 3.72607130372666437187579326886, 4.96766037348183799963978990077, 6.65604927485826000708376738925, 6.98877334052832534954356455659, 7.962315295393946388486823364661, 9.152113799421465185879419464658, 9.604068565049730737073696184901, 10.59084238742449518113113438424
