L(s) = 1 | + 2·2-s + (−5.11 − 0.917i)3-s + 4·4-s − 12.8i·5-s + (−10.2 − 1.83i)6-s − 4.12·7-s + 8·8-s + (25.3 + 9.38i)9-s − 25.7i·10-s + 61.2·11-s + (−20.4 − 3.66i)12-s + 27.5i·13-s − 8.24·14-s + (−11.8 + 65.8i)15-s + 16·16-s − 98.6i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.984 − 0.176i)3-s + 0.5·4-s − 1.15i·5-s + (−0.696 − 0.124i)6-s − 0.222·7-s + 0.353·8-s + (0.937 + 0.347i)9-s − 0.814i·10-s + 1.67·11-s + (−0.492 − 0.0882i)12-s + 0.587i·13-s − 0.157·14-s + (−0.203 + 1.13i)15-s + 0.250·16-s − 1.40i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.806114649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.806114649\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (5.11 + 0.917i)T \) |
| 59 | \( 1 + (113. - 438. i)T \) |
good | 5 | \( 1 + 12.8iT - 125T^{2} \) |
| 7 | \( 1 + 4.12T + 343T^{2} \) |
| 11 | \( 1 - 61.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 98.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 298. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 143. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 183. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 203. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 20.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 212. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 314. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 641. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 323. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.24e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 735.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 424.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3iT - 9.12e5T^{2} \) |
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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
−11.18912940199042229585208835302, −9.801291555172555076053121092919, −9.064824848808913580218609856410, −7.67906740197282948154322147291, −6.49449778121962345116915360869, −5.92905418705733192997104360377, −4.56601005264400100830776493287, −4.15787357613756777126183039318, −1.92194423882997281121776297564, −0.57728183978637523492684760054,
1.62670497821756151347034779441, 3.43420545717459178571468388085, 4.19769035769894827684197583023, 5.63882898692453541298416029481, 6.59050970228992281966412043551, 6.83075874619337316128669873606, 8.479804926782022375272959435416, 9.949466968380314814739531628685, 10.67298356485661209185385960918, 11.24592344397850704290544092082