L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 2·11-s + 12-s − 3·13-s − 14-s + 16-s + 2·17-s + 18-s − 19-s − 21-s − 2·22-s + 4·23-s + 24-s − 3·26-s + 27-s − 28-s + 4·29-s − 4·31-s + 32-s − 2·33-s + 2·34-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 − 0.603·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.218·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.982516188\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.982516188\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.66896060971342, −14.96946893984248, −14.47885907766398, −14.22570944836624, −13.33098221554130, −13.05984498984079, −12.48598143720526, −12.07211880268145, −11.24316751852394, −10.73542663466145, −10.08999441682084, −9.597561437472254, −8.954556953180045, −8.273102921915126, −7.490947259450945, −7.309955280210209, −6.416266645730464, −5.858422321665721, −5.038677006672238, −4.624727387632692, −3.792039440320369, −3.075570602383847, −2.615700433553601, −1.833732671470210, −0.6950274921012431,
0.6950274921012431, 1.833732671470210, 2.615700433553601, 3.075570602383847, 3.792039440320369, 4.624727387632692, 5.038677006672238, 5.858422321665721, 6.416266645730464, 7.309955280210209, 7.490947259450945, 8.273102921915126, 8.954556953180045, 9.597561437472254, 10.08999441682084, 10.73542663466145, 11.24316751852394, 12.07211880268145, 12.48598143720526, 13.05984498984079, 13.33098221554130, 14.22570944836624, 14.47885907766398, 14.96946893984248, 15.66896060971342