L(s) = 1 | + (1.40 + 0.117i)2-s − 3-s + (1.97 + 0.329i)4-s − i·5-s + (−1.40 − 0.117i)6-s + (0.776 − 2.52i)7-s + (2.74 + 0.695i)8-s + 9-s + (0.117 − 1.40i)10-s − 0.556i·11-s + (−1.97 − 0.329i)12-s + 0.182i·13-s + (1.38 − 3.47i)14-s + i·15-s + (3.78 + 1.30i)16-s − 2.39i·17-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0827i)2-s − 0.577·3-s + (0.986 + 0.164i)4-s − 0.447i·5-s + (−0.575 − 0.0477i)6-s + (0.293 − 0.956i)7-s + (0.969 + 0.246i)8-s + 0.333·9-s + (0.0370 − 0.445i)10-s − 0.167i·11-s + (−0.569 − 0.0952i)12-s + 0.0505i·13-s + (0.371 − 0.928i)14-s + 0.258i·15-s + (0.945 + 0.325i)16-s − 0.580i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17954 - 0.514318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17954 - 0.514318i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.117i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.776 + 2.52i)T \) |
good | 11 | \( 1 + 0.556iT - 11T^{2} \) |
| 13 | \( 1 - 0.182iT - 13T^{2} \) |
| 17 | \( 1 + 2.39iT - 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 - 3.94iT - 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 - 9.64iT - 41T^{2} \) |
| 43 | \( 1 - 0.643iT - 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 - 14.3iT - 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 1.36iT - 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 13.0iT - 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 8.01iT - 89T^{2} \) |
| 97 | \( 1 + 0.445iT - 97T^{2} \) |
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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
−11.41391082172705880039705689708, −10.50841126636632361080528119567, −9.550046934286846171798270400003, −7.992546147709042410096122818639, −7.25580133025427542805106661891, −6.25034828110398219054342082773, −5.17766083469080327742055039999, −4.45557421211286439855967002100, −3.27128715977452165964570632130, −1.36794876585859532499764946031,
1.93113602641754270590606672714, 3.22085010000612524207595116672, 4.55950608704857254266284925170, 5.51270862378189592117991957165, 6.28105984969235203903554149748, 7.23446181908725064705396784572, 8.366852541929243825684057902514, 9.754250126706708278001590876914, 10.69006431599885842863472524357, 11.41390542190745902301180395096