| L(s) = 1 | + (−3.06 − 2.57i)2-s + (−22.4 + 8.17i)3-s + (2.77 + 15.7i)4-s + (−4.34 + 24.6i)5-s + (89.7 + 32.6i)6-s + (−29.2 − 50.6i)7-s + (32.0 − 55.4i)8-s + (251. − 210. i)9-s + (76.7 − 64.4i)10-s + (251. − 436. i)11-s + (−191. − 331. i)12-s + (121. + 44.3i)13-s + (−40.5 + 230. i)14-s + (−103. − 589. i)15-s + (−240. + 87.5i)16-s + (459. + 385. i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−1.44 + 0.524i)3-s + (0.0868 + 0.492i)4-s + (−0.0778 + 0.441i)5-s + (1.01 + 0.370i)6-s + (−0.225 − 0.390i)7-s + (0.176 − 0.306i)8-s + (1.03 − 0.866i)9-s + (0.242 − 0.203i)10-s + (0.627 − 1.08i)11-s + (−0.383 − 0.663i)12-s + (0.199 + 0.0727i)13-s + (−0.0553 + 0.313i)14-s + (−0.119 − 0.676i)15-s + (−0.234 + 0.0855i)16-s + (0.385 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.555789 - 0.292878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.555789 - 0.292878i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.06 + 2.57i)T \) |
| 19 | \( 1 + (-1.14e3 + 1.08e3i)T \) |
| good | 3 | \( 1 + (22.4 - 8.17i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (4.34 - 24.6i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (29.2 + 50.6i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-251. + 436. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-121. - 44.3i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-459. - 385. i)T + (2.46e5 + 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-199. - 1.13e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (564. - 473. i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (3.61e3 + 6.26e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.47e4 + 5.36e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-3.25e3 + 1.84e4i)T + (-1.38e8 - 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-1.28e4 + 1.07e4i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + (-5.06e3 - 2.86e4i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (5.27e3 + 4.43e3i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (7.06e3 + 4.00e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (4.65e4 - 3.90e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (3.12e3 - 1.77e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + (-5.09e4 + 1.85e4i)T + (1.58e9 - 1.33e9i)T^{2} \) |
| 79 | \( 1 + (-4.39e4 + 1.60e4i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-2.92e4 - 5.07e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.49e4 - 5.43e3i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (3.16e4 + 2.65e4i)T + (1.49e9 + 8.45e9i)T^{2} \) |
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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
−15.55295768477283505516981235233, −13.77701448292447593651212246899, −12.16865633345682660340870359231, −11.16201342263505846739523298735, −10.54067730688649120347720042079, −9.126356428411551337662919293346, −7.08589020025384883971391152649, −5.66470223531863966286937345951, −3.70310621347217843021927591631, −0.63682360174963044898927458702,
1.18137513252374384393487407234, 4.99467244806506474442888177585, 6.24803834458244158194433410973, 7.41381334662894526034381496437, 9.190015618237505827297575390443, 10.57898843617472170623277794336, 11.97325691416529268242024919245, 12.62032734421672618838265298926, 14.46632991659878084412525701479, 15.98934757865579624057083176823
