L(s) = 1 | − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯ |
L(s) = 1 | − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8490923334\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8490923334\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.403900991727383454971185836199, −8.735792119741634860765562326784, −7.58930044922002904632021011796, −7.26809148398465079991793981317, −6.28936765790551897519371521509, −5.42078580743045591351431096127, −4.15390296872720246988521497798, −3.75070727266357459269629251941, −1.78027265371229092511307052832, −1.28936431270954069317179779520,
1.07026718400937328860511981678, 2.25127978319250414000309853719, 3.12570361017273950383304364602, 4.67576576351142655721747271211, 5.38510191980781384452550227640, 6.16191576010009351777978057046, 6.96484129443832487962415621855, 8.134508540993616110825691166980, 8.613284827665564718178877719368, 9.208704971631530789653962510550