| L(s) = 1 | − 2-s + 4-s + (−2.00 + 0.997i)5-s − 8-s + (2.00 − 0.997i)10-s − 0.311i·11-s + 1.62·13-s + 16-s − 1.60i·17-s + 4.54i·19-s + (−2.00 + 0.997i)20-s + 0.311i·22-s − 4.42·23-s + (3.00 − 3.99i)25-s − 1.62·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.894 + 0.446i)5-s − 0.353·8-s + (0.632 − 0.315i)10-s − 0.0939i·11-s + 0.451·13-s + 0.250·16-s − 0.390i·17-s + 1.04i·19-s + (−0.447 + 0.223i)20-s + 0.0664i·22-s − 0.922·23-s + (0.601 − 0.798i)25-s − 0.319·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3842483143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3842483143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.00 - 0.997i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 0.311iT - 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 + 1.60iT - 17T^{2} \) |
| 19 | \( 1 - 4.54iT - 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 + 1.79iT - 29T^{2} \) |
| 31 | \( 1 + 0.415iT - 31T^{2} \) |
| 37 | \( 1 - 2.16iT - 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 - 0.812iT - 43T^{2} \) |
| 47 | \( 1 - 0.316iT - 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.94iT - 61T^{2} \) |
| 67 | \( 1 + 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 2.38iT - 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 - 9.49iT - 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.033545366101451428800940231677, −7.69770854358933138053655214317, −6.73370258398090213641703237099, −6.22316535569240960827604336056, −5.25404317020908573872064156965, −4.14910561483546674813542949622, −3.51202445731441929388086748908, −2.59880950654227587524221838855, −1.47231345042610843006278090052, −0.16698276970652289691937451740,
0.960951127683438029747691998884, 2.06397346646629422942531369519, 3.20626521133852880580071044175, 3.98071618693813058958979091035, 4.82763491270023257141410568531, 5.69593881794570206560724681375, 6.64974915524779260436808503251, 7.20957116721379101700727530985, 8.096673601166200913462079711786, 8.425697865316603219964519529075
