L(s) = 1 | + (−0.331 + 7.99i)2-s + (−63.7 − 5.30i)4-s + 55.9·5-s − 552. i·7-s + (63.5 − 508. i)8-s + (−18.5 + 446. i)10-s − 1.28e3i·11-s − 2.51e3·13-s + (4.41e3 + 183. i)14-s + (4.03e3 + 676. i)16-s − 4.59e3·17-s + 9.80e3i·19-s + (−3.56e3 − 296. i)20-s + (1.02e4 + 426. i)22-s + 1.52e4i·23-s + ⋯ |
L(s) = 1 | + (−0.0414 + 0.999i)2-s + (−0.996 − 0.0829i)4-s + 0.447·5-s − 1.60i·7-s + (0.124 − 0.992i)8-s + (−0.0185 + 0.446i)10-s − 0.965i·11-s − 1.14·13-s + (1.60 + 0.0667i)14-s + (0.986 + 0.165i)16-s − 0.935·17-s + 1.42i·19-s + (−0.445 − 0.0370i)20-s + (0.964 + 0.0400i)22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0829i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4918880351\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4918880351\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.331 - 7.99i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 55.9T \) |
good | 7 | \( 1 + 552. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.28e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.51e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.59e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.80e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.52e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.43e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.44e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.14e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.76e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.53e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.31e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.28e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.71e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.52e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 7.17e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.93e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.25e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.09e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.92e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.54e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.46e6T + 8.32e11T^{2} \) |
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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
−12.26270353127708440395534202920, −10.67721181895801597279070216337, −10.02496355799948910729299937703, −8.889498792075824399007718982217, −7.71252887845843706990885768790, −6.96888804755442014510566387515, −5.84354858436625470996352015923, −4.64057194272119529286751377526, −3.50229015731722834443895208929, −1.21608680609291343984198053208,
0.14737194317853814818079906570, 2.19956332938152115509407297180, 2.53126207628334194652835346368, 4.54802594375155182030211737008, 5.34167696787587725604973218288, 6.84117434966063597286198262729, 8.492561625641401116506727731936, 9.229093454967639842321667677754, 10.00342598624549626193104325207, 11.18579291211447934327828917587