L(s) = 1 | + (3.66e4 − 3.66e4i)2-s + (1.35e8 + 1.35e8i)3-s + 1.44e10i·4-s + (6.50e11 + 3.99e11i)5-s + 9.95e12·6-s + (−1.47e14 + 1.47e14i)7-s + (1.16e15 + 1.16e15i)8-s + 2.02e16i·9-s + (3.84e16 − 9.20e15i)10-s + 7.69e17·11-s + (−1.96e18 + 1.96e18i)12-s + (5.59e18 + 5.59e18i)13-s + 1.08e19i·14-s + (3.41e19 + 1.42e20i)15-s − 1.64e20·16-s + (−4.36e19 + 4.36e19i)17-s + ⋯ |
L(s) = 1 | + (0.279 − 0.279i)2-s + (1.05 + 1.05i)3-s + 0.843i·4-s + (0.852 + 0.522i)5-s + 0.588·6-s + (−0.635 + 0.635i)7-s + (0.515 + 0.515i)8-s + 1.21i·9-s + (0.384 − 0.0920i)10-s + 1.52·11-s + (−0.887 + 0.887i)12-s + (0.646 + 0.646i)13-s + 0.355i·14-s + (0.346 + 1.44i)15-s − 0.555·16-s + (−0.0527 + 0.0527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{35}{2})\) |
\(\approx\) |
\(4.167083442\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.167083442\) |
\(L(18)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-6.50e11 - 3.99e11i)T \) |
good | 2 | \( 1 + (-3.66e4 + 3.66e4i)T - 1.71e10iT^{2} \) |
| 3 | \( 1 + (-1.35e8 - 1.35e8i)T + 1.66e16iT^{2} \) |
| 7 | \( 1 + (1.47e14 - 1.47e14i)T - 5.41e28iT^{2} \) |
| 11 | \( 1 - 7.69e17T + 2.55e35T^{2} \) |
| 13 | \( 1 + (-5.59e18 - 5.59e18i)T + 7.48e37iT^{2} \) |
| 17 | \( 1 + (4.36e19 - 4.36e19i)T - 6.84e41iT^{2} \) |
| 19 | \( 1 + 9.91e21iT - 3.00e43T^{2} \) |
| 23 | \( 1 + (4.35e22 + 4.35e22i)T + 1.98e46iT^{2} \) |
| 29 | \( 1 + 1.16e25iT - 5.26e49T^{2} \) |
| 31 | \( 1 + 6.82e24T + 5.08e50T^{2} \) |
| 37 | \( 1 + (3.61e25 - 3.61e25i)T - 2.08e53iT^{2} \) |
| 41 | \( 1 + 1.34e27T + 6.83e54T^{2} \) |
| 43 | \( 1 + (-2.87e27 - 2.87e27i)T + 3.45e55iT^{2} \) |
| 47 | \( 1 + (6.80e27 - 6.80e27i)T - 7.10e56iT^{2} \) |
| 53 | \( 1 + (7.11e28 + 7.11e28i)T + 4.22e58iT^{2} \) |
| 59 | \( 1 + 1.99e30iT - 1.61e60T^{2} \) |
| 61 | \( 1 - 4.23e30T + 5.02e60T^{2} \) |
| 67 | \( 1 + (1.01e31 - 1.01e31i)T - 1.22e62iT^{2} \) |
| 71 | \( 1 - 8.91e30T + 8.76e62T^{2} \) |
| 73 | \( 1 + (2.78e31 + 2.78e31i)T + 2.25e63iT^{2} \) |
| 79 | \( 1 + 3.78e31iT - 3.30e64T^{2} \) |
| 83 | \( 1 + (5.24e32 + 5.24e32i)T + 1.77e65iT^{2} \) |
| 89 | \( 1 - 1.16e33iT - 1.90e66T^{2} \) |
| 97 | \( 1 + (2.55e33 - 2.55e33i)T - 3.55e67iT^{2} \) |
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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
−15.87441091771226815929963693051, −14.43022019105782039811639570866, −13.34278515380206332547192793858, −11.42569332467883811801987374436, −9.537000705239615592226791424049, −8.796040026391033527992860110785, −6.55555168961117172484266434676, −4.29693630916752765037966904940, −3.20060771764331979102325102126, −2.20831228597541000310980383218,
1.07304075959914431616357344694, 1.66472152869936173216043344066, 3.67589324937944864231590940539, 5.86971573821426713614924314721, 6.96048291759899756716938765360, 8.780482269279880013476529263090, 10.08227937193802382100853298153, 12.68780408923338556811670286731, 13.79287408266869202493934832340, 14.47757466232178515066998797703