L(s) = 1 | + (−1.41 − i)3-s + (−1.73 − 1.41i)5-s + (1.00 + 2.82i)9-s + 4.89·11-s + 4.89i·13-s + (1.03 + 3.73i)15-s + 3.46·17-s − 3.46i·19-s − 6i·23-s + (0.999 + 4.89i)25-s + (1.41 − 5.00i)27-s + 2.82i·29-s − 3.46i·31-s + (−6.92 − 4.89i)33-s − 4.89i·37-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + (−0.774 − 0.632i)5-s + (0.333 + 0.942i)9-s + 1.47·11-s + 1.35i·13-s + (0.267 + 0.963i)15-s + 0.840·17-s − 0.794i·19-s − 1.25i·23-s + (0.199 + 0.979i)25-s + (0.272 − 0.962i)27-s + 0.525i·29-s − 0.622i·31-s + (−1.20 − 0.852i)33-s − 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.839696 - 0.638473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.839696 - 0.638473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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
−9.796201861167554997712626506840, −8.923411037309557154584054800469, −8.216487314124212075934696738303, −6.96764112111054297150292911377, −6.75199446757046058570642341558, −5.48855995132299066541149026207, −4.54626987139314436144150564894, −3.79069980642322463378539708676, −1.92032048227351568161557389131, −0.70878724109305976130769868954,
1.10590715720127741405859031475, 3.31681626728289923267225395437, 3.76204759362390188156028090529, 4.96580992860699539275014773941, 5.93226855493986495795362313597, 6.67460727394738789253465605140, 7.63692694719280745851624313993, 8.471932206611649845092382850840, 9.752501364489308777753022608925, 10.11140995973600589384787601112