| L(s) = 1 | + (−32 − 55.4i)2-s + (−513 + 888. i)3-s + (−2.04e3 + 3.54e3i)4-s + (2.16e3 + 3.74e3i)5-s + 6.56e4·6-s + 2.62e5·8-s + (2.70e5 + 4.69e5i)9-s + (1.38e5 − 2.39e5i)10-s + (4.39e6 − 7.61e6i)11-s + (−2.10e6 − 3.63e6i)12-s + 2.04e7·13-s − 4.43e6·15-s + (−8.38e6 − 1.45e7i)16-s + (8.59e5 − 1.48e6i)17-s + (1.73e7 − 3.00e7i)18-s + (−5.48e7 − 9.50e7i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.406 + 0.703i)3-s + (−0.249 + 0.433i)4-s + (0.0618 + 0.107i)5-s + 0.574·6-s + 0.353·8-s + (0.169 + 0.294i)9-s + (0.0437 − 0.0757i)10-s + (0.747 − 1.29i)11-s + (−0.203 − 0.351i)12-s + 1.17·13-s − 0.100·15-s + (−0.125 − 0.216i)16-s + (0.00863 − 0.0149i)17-s + (0.120 − 0.208i)18-s + (−0.267 − 0.463i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.418073334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418073334\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (32 + 55.4i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (513 - 888. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-2.16e3 - 3.74e3i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-4.39e6 + 7.61e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 - 2.04e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-8.59e5 + 1.48e6i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (5.48e7 + 9.50e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-3.23e8 - 5.60e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 - 7.28e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-5.14e8 + 8.90e8i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (7.11e9 + 1.23e10i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 4.45e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.46e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-2.39e10 - 4.14e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-8.49e10 + 1.47e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (1.50e11 - 2.60e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-1.84e11 - 3.20e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.93e11 + 6.81e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 - 5.59e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-6.05e10 + 1.04e11i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (1.45e11 + 2.51e11i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 - 3.96e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (3.01e12 + 5.21e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + 1.13e13T + 6.73e25T^{2} \) |
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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
−11.04347210025940194188550659564, −10.31227065174134196231315541456, −9.100972738243814031760981522617, −8.276091784516939875750192602495, −6.64357774131558038604454100453, −5.40751852382742602106715316276, −4.10126480337009116614034836887, −3.18790515163779347031753690166, −1.58946161854468050858559789255, −0.44108690700528761403665989144,
0.995448120222954595862324476649, 1.73001184347645447569747961200, 3.76605930041484420798162292126, 5.10275559847559588667890028124, 6.50799051055870502888176635863, 6.88511939718976379848072479054, 8.253195853930942092376106261060, 9.289033583800896213528401013880, 10.38741913924268999053597143186, 11.71214800219690408380829736369
