| L(s) = 1 | + 3-s + 9i·5-s − 15i·7-s − 26·9-s − 48i·11-s + (−26 + 39i)13-s + 9i·15-s − 45·17-s − 6i·19-s − 15i·21-s + 162·23-s + 44·25-s − 53·27-s + 144·29-s + 264i·31-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 0.804i·5-s − 0.809i·7-s − 0.962·9-s − 1.31i·11-s + (−0.554 + 0.832i)13-s + 0.154i·15-s − 0.642·17-s − 0.0724i·19-s − 0.155i·21-s + 1.46·23-s + 0.351·25-s − 0.377·27-s + 0.922·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.613176289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613176289\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (26 - 39i)T \) |
| good | 3 | \( 1 - T + 27T^{2} \) |
| 5 | \( 1 - 9iT - 125T^{2} \) |
| 7 | \( 1 + 15iT - 343T^{2} \) |
| 11 | \( 1 + 48iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 45T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 162T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144T + 2.43e4T^{2} \) |
| 31 | \( 1 - 264iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 303iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 192iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 97T + 7.95e4T^{2} \) |
| 47 | \( 1 + 111iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 414T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 376T + 2.26e5T^{2} \) |
| 67 | \( 1 - 36iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 357iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 830T + 4.93e5T^{2} \) |
| 83 | \( 1 - 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 438iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 852iT - 9.12e5T^{2} \) |
| 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
−10.12296809073453946577101063611, −8.844093072386562623548337158814, −8.521505677276996807502124226748, −7.10322561178696918867101861513, −6.76441031312008920036763712589, −5.61644359095244283845608616914, −4.50899804948571491161300413255, −3.27982842270489318115305124450, −2.67158487467667240152009903124, −0.934227055581934061885960392950,
0.51072389304243282912492477681, 2.12410203811118891024242691836, 2.94272419001192202638805585880, 4.49737409637785754147429946039, 5.17002361724921279450388141534, 6.04328200249183860735939913211, 7.24809692081464658043496853640, 8.103220612437110596306737915551, 8.993850620292669177407771453071, 9.377484837846237009425259684474
