| L(s) = 1 | + (−9.99 + 25.0i)3-s − 134. i·7-s + (−529. − 501. i)9-s + 1.15e3i·11-s − 78.2i·13-s + 54.6·17-s − 1.98e3·19-s + (3.37e3 + 1.34e3i)21-s − 1.69e4·23-s + (1.78e4 − 8.27e3i)27-s − 1.84e4i·29-s + 3.83e4·31-s + (−2.89e4 − 1.15e4i)33-s − 2.12e4i·37-s + (1.96e3 + 781. i)39-s + ⋯ |
| L(s) = 1 | + (−0.370 + 0.929i)3-s − 0.392i·7-s + (−0.726 − 0.687i)9-s + 0.867i·11-s − 0.0356i·13-s + 0.0111·17-s − 0.289·19-s + (0.364 + 0.145i)21-s − 1.39·23-s + (0.907 − 0.420i)27-s − 0.757i·29-s + 1.28·31-s + (−0.806 − 0.321i)33-s − 0.419i·37-s + (0.0330 + 0.0131i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0845i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.996 - 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.380154819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380154819\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (9.99 - 25.0i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 134. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.15e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 78.2iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 54.6T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.98e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.69e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.84e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.83e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.12e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 8.89e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.55e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.63e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 9.71e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.86e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.19e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.42e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.63e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.76e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 6.27e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.91e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 8.27e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 8.48e5iT - 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43102994916602941680264686449, −10.04872500788719019078997811623, −9.003678568331355359556611243250, −7.909489713283217483774685959208, −6.70023385194768846757466207089, −5.65680163207686790698405633474, −4.52603996450092769402126548760, −3.77405721511974649062889299622, −2.24811404770081743863447681490, −0.49936037512200265300275375172,
0.75812283013459947379916580757, 2.00548775509511783327493241636, 3.20909237576291098757858211014, 4.83770321314459226292777593668, 5.98445529438459039439821189017, 6.60011796690852531270841636055, 7.949531506509839634889266476537, 8.478492135111363195909462852298, 9.792918084271302298122843979019, 10.92952297355637635776275322291
