L(s) = 1 | − 5-s − 2.08i·7-s − 4.49·11-s + 0.851·13-s − 1.37·17-s − 4.41i·19-s + (0.597 − 4.75i)23-s + 25-s − 9.76i·29-s − 8.80·31-s + 2.08i·35-s + 8.90i·37-s − 10.5i·41-s + 9.03i·43-s − 3.43i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.789i·7-s − 1.35·11-s + 0.236·13-s − 0.333·17-s − 1.01i·19-s + (0.124 − 0.992i)23-s + 0.200·25-s − 1.81i·29-s − 1.58·31-s + 0.352i·35-s + 1.46i·37-s − 1.64i·41-s + 1.37i·43-s − 0.501i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06580682230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06580682230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-0.597 + 4.75i)T \) |
good | 7 | \( 1 + 2.08iT - 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 0.851T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 4.41iT - 19T^{2} \) |
| 29 | \( 1 + 9.76iT - 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 8.90iT - 37T^{2} \) |
| 41 | \( 1 + 10.5iT - 41T^{2} \) |
| 43 | \( 1 - 9.03iT - 43T^{2} \) |
| 47 | \( 1 + 3.43iT - 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 - 1.51iT - 59T^{2} \) |
| 61 | \( 1 - 5.74iT - 61T^{2} \) |
| 67 | \( 1 - 3.63iT - 67T^{2} \) |
| 71 | \( 1 + 0.419iT - 71T^{2} \) |
| 73 | \( 1 - 5.12T + 73T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 3.95T + 89T^{2} \) |
| 97 | \( 1 - 4.55iT - 97T^{2} \) |
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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
−8.060574124062270706204247575960, −7.26317066464808127524394935214, −6.89818273758890326705291627241, −5.93392253696618998238163389047, −5.18887782890171937036108974787, −4.44070251552176850079988006168, −3.89266934072166401752511364417, −2.86668955465489469775719044102, −2.23657018636657360328151453296, −0.828028760694952309264801573027,
0.01887682100330163993181622906, 1.54630192393422899257962797847, 2.37856116321835837525392971910, 3.29057071179044348506881342835, 3.85934655789964027226326076453, 5.01346024169134759528845138432, 5.42653514459689928241551883717, 6.03980630452984792114591120627, 7.08941143187051928688283006846, 7.57775819713404483194434313226