L(s) = 1 | + (1.12e5 + 5.12e5i)2-s − 1.96e9i·3-s + (−2.49e11 + 1.15e11i)4-s − 3.09e13·5-s + (1.00e15 − 2.21e14i)6-s + 9.14e15i·7-s + (−8.71e16 − 1.14e17i)8-s − 2.50e18·9-s + (−3.48e18 − 1.58e19i)10-s + 4.11e19i·11-s + (2.26e20 + 4.89e20i)12-s − 1.68e21·13-s + (−4.68e21 + 1.02e21i)14-s + 6.07e22i·15-s + (4.89e22 − 5.75e22i)16-s + 6.20e22·17-s + ⋯ |
L(s) = 1 | + (0.214 + 0.976i)2-s − 1.68i·3-s + (−0.907 + 0.419i)4-s − 1.62·5-s + (1.64 − 0.362i)6-s + 0.802i·7-s + (−0.604 − 0.796i)8-s − 1.85·9-s + (−0.348 − 1.58i)10-s + 0.673i·11-s + (0.708 + 1.53i)12-s − 1.15·13-s + (−0.783 + 0.172i)14-s + 2.74i·15-s + (0.648 − 0.761i)16-s + 0.259·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{39}{2})\) |
\(\approx\) |
\(0.8254847808\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8254847808\) |
\(L(20)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.12e5 - 5.12e5i)T \) |
good | 3 | \( 1 + 1.96e9iT - 1.35e18T^{2} \) |
| 5 | \( 1 + 3.09e13T + 3.63e26T^{2} \) |
| 7 | \( 1 - 9.14e15iT - 1.29e32T^{2} \) |
| 11 | \( 1 - 4.11e19iT - 3.74e39T^{2} \) |
| 13 | \( 1 + 1.68e21T + 2.13e42T^{2} \) |
| 17 | \( 1 - 6.20e22T + 5.71e46T^{2} \) |
| 19 | \( 1 + 1.64e24iT - 3.91e48T^{2} \) |
| 23 | \( 1 + 7.68e25iT - 5.56e51T^{2} \) |
| 29 | \( 1 - 9.19e27T + 3.72e55T^{2} \) |
| 31 | \( 1 - 1.50e28iT - 4.69e56T^{2} \) |
| 37 | \( 1 + 3.34e29T + 3.90e59T^{2} \) |
| 41 | \( 1 + 4.03e29T + 1.93e61T^{2} \) |
| 43 | \( 1 + 9.22e30iT - 1.17e62T^{2} \) |
| 47 | \( 1 - 8.34e31iT - 3.46e63T^{2} \) |
| 53 | \( 1 + 8.95e32T + 3.33e65T^{2} \) |
| 59 | \( 1 - 2.84e33iT - 1.96e67T^{2} \) |
| 61 | \( 1 - 5.02e33T + 6.95e67T^{2} \) |
| 67 | \( 1 - 3.32e34iT - 2.45e69T^{2} \) |
| 71 | \( 1 + 1.07e33iT - 2.22e70T^{2} \) |
| 73 | \( 1 - 4.49e35T + 6.40e70T^{2} \) |
| 79 | \( 1 - 2.34e35iT - 1.28e72T^{2} \) |
| 83 | \( 1 + 3.08e36iT - 8.41e72T^{2} \) |
| 89 | \( 1 + 4.93e36T + 1.19e74T^{2} \) |
| 97 | \( 1 - 3.62e37T + 3.14e75T^{2} \) |
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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
−15.66587065856079882104760851888, −14.42116413194391584877181738030, −12.51907957596018543892089256747, −12.08044381424259670324378044111, −8.568316908779569397621223574447, −7.55328541070358570954674976768, −6.69721186361533398468756644039, −4.79440512117220800677472371498, −2.74255685999802405772369108960, −0.57910890815766653152658628481,
0.45802665497116310681613719135, 3.24721192773905434013514303850, 3.96616256373016499995854797447, 4.97612616504818042478518286326, 8.132036933423025032586157120342, 9.787982447741362681025447328676, 10.89569063260163096214485872103, 11.95364255585572154876776023638, 14.29612556513852962452957024927, 15.51434441750371754240118723281