L(s) = 1 | + (−0.274 + 2.21i)5-s + 0.615i·7-s − 5.98·11-s − 1.31i·13-s − 5.22i·17-s + 6.47·19-s − i·23-s + (−4.84 − 1.21i)25-s − 3.81·29-s + 2.89·31-s + (−1.36 − 0.168i)35-s + 3.05i·37-s + 8.67·41-s + 2.78i·43-s − 6.52i·47-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.992i)5-s + 0.232i·7-s − 1.80·11-s − 0.363i·13-s − 1.26i·17-s + 1.48·19-s − 0.208i·23-s + (−0.969 − 0.243i)25-s − 0.707·29-s + 0.520·31-s + (−0.230 − 0.0285i)35-s + 0.502i·37-s + 1.35·41-s + 0.425i·43-s − 0.952i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.418929458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418929458\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.274 - 2.21i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 - 0.615iT - 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 + 1.31iT - 13T^{2} \) |
| 17 | \( 1 + 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 - 3.05iT - 37T^{2} \) |
| 41 | \( 1 - 8.67T + 41T^{2} \) |
| 43 | \( 1 - 2.78iT - 43T^{2} \) |
| 47 | \( 1 + 6.52iT - 47T^{2} \) |
| 53 | \( 1 + 8.11iT - 53T^{2} \) |
| 59 | \( 1 + 8.93T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 16.1iT - 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 + 7.29iT - 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−8.136929958004775194406858328404, −7.61509155753236741548417090919, −7.16177789464775555119872863540, −6.16867288210316269578841802067, −5.36720005857600680646689997117, −4.87820450380326963549164242838, −3.54706192192137768188474328482, −2.85376488432096204190763422579, −2.29448230698426217065262368126, −0.55451446842482385777166395572,
0.794061410162024236443317665420, 1.90605128093553497208077867162, 2.95783948547140838656824338252, 3.96641293856217719734330967762, 4.69948854567293010124658288771, 5.52512599952681445070531442827, 5.91834655820969250069431479765, 7.24538424226029362352674979508, 7.80301544220234974894732779958, 8.249460053709770079857057372493