L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 2.03·5-s − 0.999·8-s + (−1.01 − 1.75i)10-s − 4·11-s + (2.12 + 3.67i)13-s + (−0.5 − 0.866i)16-s + (0.707 + 1.22i)17-s + (0.398 − 0.690i)19-s + (1.01 − 1.75i)20-s + (−2 − 3.46i)22-s − 6.74·23-s − 0.872·25-s + (−2.12 + 3.67i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.908·5-s − 0.353·8-s + (−0.321 − 0.556i)10-s − 1.20·11-s + (0.588 + 1.01i)13-s + (−0.125 − 0.216i)16-s + (0.171 + 0.297i)17-s + (0.0914 − 0.158i)19-s + (0.227 − 0.393i)20-s + (−0.426 − 0.738i)22-s − 1.40·23-s − 0.174·25-s + (−0.416 + 0.720i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7638947640\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7638947640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.03T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (-2.12 - 3.67i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.398 + 0.690i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + (-4.43 + 7.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.73 + 4.74i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.87 + 8.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.43 - 7.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.44 - 5.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.30 + 9.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.32 + 2.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.398 + 0.690i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.436 - 0.756i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 + (7.68 + 13.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.93 + 5.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.15 + 7.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.92 + 15.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.353880345909144581190755271908, −7.935550626701999116810958780453, −7.43693441589443550778057089471, −6.23695607490035819511232414935, −5.92222134978643757851395621790, −4.55996754335371110265987541313, −4.25569810971528458662480375049, −3.22605811437590871648727462211, −2.11821391935718285606820694945, −0.25444873805215802247227216601,
1.04934126525059008811751989069, 2.51015281119501595963112494287, 3.28718134838937234330235257830, 4.03574862308604400112838344251, 5.03225963146508036724302054903, 5.60922562217708273001988168476, 6.63289207179348900556388824262, 7.68062430234393041462705595310, 8.154855922185027637465379386204, 8.848187222799957900829272892723