L(s) = 1 | + (0.820 + 1.15i)2-s + (−1.69 + 0.378i)3-s + (−0.654 + 1.88i)4-s + (1.97 + 1.14i)5-s + (−1.82 − 1.63i)6-s + (−0.907 − 1.57i)7-s + (−2.71 + 0.795i)8-s + (2.71 − 1.27i)9-s + (0.306 + 3.21i)10-s + (4.24 − 2.44i)11-s + (0.391 − 3.44i)12-s + (−4.00 − 2.31i)13-s + (1.06 − 2.33i)14-s + (−3.77 − 1.18i)15-s + (−3.14 − 2.47i)16-s + 1.92·17-s + ⋯ |
L(s) = 1 | + (0.579 + 0.814i)2-s + (−0.975 + 0.218i)3-s + (−0.327 + 0.944i)4-s + (0.883 + 0.510i)5-s + (−0.743 − 0.668i)6-s + (−0.343 − 0.594i)7-s + (−0.959 + 0.281i)8-s + (0.904 − 0.426i)9-s + (0.0968 + 1.01i)10-s + (1.27 − 0.738i)11-s + (0.113 − 0.993i)12-s + (−1.11 − 0.641i)13-s + (0.285 − 0.624i)14-s + (−0.973 − 0.304i)15-s + (−0.785 − 0.618i)16-s + 0.467·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.748387 + 0.645552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.748387 + 0.645552i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.820 - 1.15i)T \) |
| 3 | \( 1 + (1.69 - 0.378i)T \) |
good | 5 | \( 1 + (-1.97 - 1.14i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.907 + 1.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.24 + 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.00 + 2.31i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 2.12iT - 19T^{2} \) |
| 23 | \( 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.98iT - 37T^{2} \) |
| 41 | \( 1 + (2.36 - 4.09i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.02 + 3.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.95iT - 53T^{2} \) |
| 59 | \( 1 + (3.05 + 1.76i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.71 - 0.991i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.72 - 4.46i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (4.97 + 8.61i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.12 - 1.80i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.49T + 89T^{2} \) |
| 97 | \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.73608679053032739279642951057, −14.02366912812648035145857858329, −12.81542388660747914613891798513, −11.81487156421440477638811344470, −10.42154244715590256735227066772, −9.351844746094888919264930361934, −7.35967194995872073588417269346, −6.34277869647692455373209935863, −5.41689759604860908302065866629, −3.71141571055986324987796352922,
1.89339069970918461618636920027, 4.50058612110268657776741812499, 5.64880768670881203338771401433, 6.75133116906701408665026130120, 9.377854038708136727060839329173, 9.852559615536150874796920553846, 11.46167129406626415193024478684, 12.22869471551045459314014241281, 12.95129972740286132079918338835, 14.15363317861161169866121288324