L(s) = 1 | + (0.532 − 0.846i)3-s + (−0.781 + 0.623i)4-s + (−0.943 + 0.330i)7-s + (−0.433 − 0.900i)9-s + (0.111 + 0.993i)12-s + (−0.146 − 0.420i)13-s + (0.222 − 0.974i)16-s + 0.867·19-s + (−0.222 + 0.974i)21-s + (−0.993 − 0.111i)27-s + (0.532 − 0.846i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.55 + 0.175i)37-s + (−0.433 − 0.0990i)39-s + ⋯ |
L(s) = 1 | + (0.532 − 0.846i)3-s + (−0.781 + 0.623i)4-s + (−0.943 + 0.330i)7-s + (−0.433 − 0.900i)9-s + (0.111 + 0.993i)12-s + (−0.146 − 0.420i)13-s + (0.222 − 0.974i)16-s + 0.867·19-s + (−0.222 + 0.974i)21-s + (−0.993 − 0.111i)27-s + (0.532 − 0.846i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.55 + 0.175i)37-s + (−0.433 − 0.0990i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7067030959\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7067030959\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.532 + 0.846i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.943 - 0.330i)T \) |
good | 2 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (0.146 + 0.420i)T + (-0.781 + 0.623i)T^{2} \) |
| 17 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 19 | \( 1 - 0.867T + T^{2} \) |
| 23 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 29 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + 1.56iT - T^{2} \) |
| 37 | \( 1 + (1.55 - 0.175i)T + (0.974 - 0.222i)T^{2} \) |
| 41 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.461 + 0.734i)T + (-0.433 + 0.900i)T^{2} \) |
| 47 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 59 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 + (1.37 + 1.37i)T + iT^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (1.17 + 0.411i)T + (0.781 + 0.623i)T^{2} \) |
| 79 | \( 1 + 1.24iT - T^{2} \) |
| 83 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 89 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (-1.27 + 1.27i)T - iT^{2} \) |
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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
−8.512443406028146650106326948946, −7.62904329568574339856689946169, −7.26954492583483279728093473351, −6.25874831448387742602832609502, −5.59200056296400200684890219534, −4.55989293658825024485364325809, −3.36537915832786174539287742105, −3.17987874116460189989861208854, −1.97576096905418113196058318914, −0.39089081646855797356337877290,
1.45235807419803501004625709180, 2.88336470718647220378239637265, 3.58602167205224706761199158483, 4.36886450424124535642032236849, 5.09868959606504079398628153317, 5.78562867457717566329939724617, 6.74839597229506266537618098329, 7.53841179762748853056847647206, 8.589894080073800327515205112582, 9.034492519860816883052872316078