L(s) = 1 | + (3.53 + 3.53i)5-s − 5i·7-s + 1.41i·11-s + 9i·13-s − 11.3·17-s − 21·19-s + 1.41·23-s + 25.0i·25-s − 38.1i·29-s − 40·31-s + (17.6 − 17.6i)35-s − 25i·37-s − 52.3i·41-s − 64i·43-s + 22.6·47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s − 0.714i·7-s + 0.128i·11-s + 0.692i·13-s − 0.665·17-s − 1.10·19-s + 0.0614·23-s + 1.00i·25-s − 1.31i·29-s − 1.29·31-s + (0.505 − 0.505i)35-s − 0.675i·37-s − 1.27i·41-s − 1.48i·43-s + 0.481·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5954977003\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5954977003\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.53 - 3.53i)T \) |
good | 7 | \( 1 + 5iT - 49T^{2} \) |
| 11 | \( 1 - 1.41iT - 121T^{2} \) |
| 13 | \( 1 - 9iT - 169T^{2} \) |
| 17 | \( 1 + 11.3T + 289T^{2} \) |
| 19 | \( 1 + 21T + 361T^{2} \) |
| 23 | \( 1 - 1.41T + 529T^{2} \) |
| 29 | \( 1 + 38.1iT - 841T^{2} \) |
| 31 | \( 1 + 40T + 961T^{2} \) |
| 37 | \( 1 + 25iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 22.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 72.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 97T + 3.72e3T^{2} \) |
| 67 | \( 1 - 131iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 89.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 117T + 6.24e3T^{2} \) |
| 83 | \( 1 + 57.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 147. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41iT - 9.40e3T^{2} \) |
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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
−8.852388189139691522678511801231, −7.57683409115659473416068047037, −7.06496771434930337393498562400, −6.30668597144511230388065753821, −5.58456596733105015064515099083, −4.38001022824450552800813770398, −3.77160875468792456922303715010, −2.46469806440888987275187681810, −1.76894622033294934605330827258, −0.13508166798981376824091877532,
1.35324049645282745087552969738, 2.29652701738692474314126576570, 3.28914110157508641836244195694, 4.58682408313203652006964305614, 5.17080828204824696250574661241, 6.06068095944580790446690847324, 6.60076542143041766899026620553, 7.86074355653614687308563263222, 8.499458238294057053171123929856, 9.176410235074926734081731739691