L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s − 13-s + 14-s + 16-s − 3·17-s + 18-s + 5·19-s + 21-s − 22-s − 23-s + 24-s − 5·25-s − 26-s + 27-s + 28-s + 9·29-s + 5·31-s + 32-s − 33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.14·19-s + 0.218·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.898·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.627329645\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.627329645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.24728484424496, −15.85386893038016, −15.33669747458792, −14.86132080871449, −14.08615680755041, −13.72174322812281, −13.38603173152340, −12.53279441214428, −11.93063141822093, −11.57745138151959, −10.75408912320511, −10.06592114938977, −9.672245349636039, −8.680643897831465, −8.241068255755017, −7.527527696534093, −6.972623673265869, −6.224464619159429, −5.468047504290907, −4.759062909937910, −4.223123487837712, −3.364442000437412, −2.646534334593591, −1.991544996202948, −0.8857359144973531,
0.8857359144973531, 1.991544996202948, 2.646534334593591, 3.364442000437412, 4.223123487837712, 4.759062909937910, 5.468047504290907, 6.224464619159429, 6.972623673265869, 7.527527696534093, 8.241068255755017, 8.680643897831465, 9.672245349636039, 10.06592114938977, 10.75408912320511, 11.57745138151959, 11.93063141822093, 12.53279441214428, 13.38603173152340, 13.72174322812281, 14.08615680755041, 14.86132080871449, 15.33669747458792, 15.85386893038016, 16.24728484424496