L(s) = 1 | + (−1.22 − 1.22i)3-s + (2.67 − 4.22i)5-s + (5.44 − 5.44i)7-s + 2.99i·9-s − 6.44·11-s + (−14.4 − 14.4i)13-s + (−8.44 + 1.89i)15-s + (−23.1 + 23.1i)17-s − 16.6i·19-s − 13.3·21-s + (6.65 + 6.65i)23-s + (−10.6 − 22.5i)25-s + (3.67 − 3.67i)27-s − 0.0454i·29-s − 4.49·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.534 − 0.844i)5-s + (0.778 − 0.778i)7-s + 0.333i·9-s − 0.586·11-s + (−1.11 − 1.11i)13-s + (−0.563 + 0.126i)15-s + (−1.36 + 1.36i)17-s − 0.878i·19-s − 0.635·21-s + (0.289 + 0.289i)23-s + (−0.427 − 0.903i)25-s + (0.136 − 0.136i)27-s − 0.00156i·29-s − 0.144·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7131971882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7131971882\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (-2.67 + 4.22i)T \) |
good | 7 | \( 1 + (-5.44 + 5.44i)T - 49iT^{2} \) |
| 11 | \( 1 + 6.44T + 121T^{2} \) |
| 13 | \( 1 + (14.4 + 14.4i)T + 169iT^{2} \) |
| 17 | \( 1 + (23.1 - 23.1i)T - 289iT^{2} \) |
| 19 | \( 1 + 16.6iT - 361T^{2} \) |
| 23 | \( 1 + (-6.65 - 6.65i)T + 529iT^{2} \) |
| 29 | \( 1 + 0.0454iT - 841T^{2} \) |
| 31 | \( 1 + 4.49T + 961T^{2} \) |
| 37 | \( 1 + (35.3 - 35.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 20.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.2 - 32.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-50.5 + 50.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-5.50 - 5.50i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 55.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (85.2 - 85.2i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (21.9 + 21.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-94.9 - 94.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 71.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (37 - 37i)T - 9.40e3iT^{2} \) |
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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
−9.341472063733028291348474014756, −8.404605295018649889849327394934, −7.73890714195160876278174699443, −6.87313743917224621445506669599, −5.76387442748203033417374586610, −4.98299564845268455636049344152, −4.31789454921275681416039790155, −2.56601886145525375405575141590, −1.45471275552254033395565932657, −0.22620259798419129210742561630,
2.02935951246742552899688879678, 2.70120782084679589036470415617, 4.28740757611391982401596858377, 5.10762439735251854936585994949, 5.88614360761240221297876911187, 6.93326189816607905723553299556, 7.56426733148082830168985256655, 8.992635863019533330165695225083, 9.341592105312155263906714485495, 10.44074208786141801299510345720