| L(s) = 1 | + 1.58i·3-s + (−2.09 − 0.787i)5-s − 2.84i·7-s + 0.493·9-s − 1.98·11-s + 4.69i·13-s + (1.24 − 3.31i)15-s − 3.16i·17-s − 6.16·19-s + 4.50·21-s − i·23-s + (3.75 + 3.29i)25-s + 5.53i·27-s + 6.61·29-s + 8.29·31-s + ⋯ |
| L(s) = 1 | + 0.914i·3-s + (−0.935 − 0.352i)5-s − 1.07i·7-s + 0.164·9-s − 0.598·11-s + 1.30i·13-s + (0.321 − 0.855i)15-s − 0.768i·17-s − 1.41·19-s + 0.983·21-s − 0.208i·23-s + (0.751 + 0.659i)25-s + 1.06i·27-s + 1.22·29-s + 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.226336120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226336120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.09 + 0.787i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.58iT - 3T^{2} \) |
| 7 | \( 1 + 2.84iT - 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 1.71iT - 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 0.177iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 6.18iT - 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 3.07iT - 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 3.49iT - 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.00iT - 97T^{2} \) |
| 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
−9.438564810172515905373472658046, −8.671163568571056695517530267519, −7.86218958070214344266674018163, −7.09242179955143193906984004938, −6.36547374195580167297471995646, −4.78373536936326988097426697089, −4.48034072279631922971709918781, −3.90785870242565865306017426341, −2.69012812679907104865176069013, −0.979921711663664490485275325013,
0.58066761152066023840967642971, 2.16656295679753781805609161575, 2.90049843524368639772790161748, 4.06611717425992990892327553540, 5.09575219970515337461361715937, 6.12226216820611270604197818324, 6.68048257914758044826623696541, 7.69750800033804284708744043235, 8.226231506155752257429854987594, 8.656960504134717529297597354225
