| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 7-s − 3·8-s + 9-s + 10-s + 12-s + 6·13-s + 14-s − 15-s − 16-s − 2·17-s + 18-s − 8·19-s − 20-s − 21-s − 8·23-s + 3·24-s + 25-s + 6·26-s − 27-s − 28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s − 0.218·21-s − 1.66·23-s + 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100905 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100905 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366056146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366056146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.66119483827873, −13.34481545087183, −12.83513470245134, −12.39502350703085, −11.85804327172355, −11.39222087775162, −10.77397207789587, −10.47140372846643, −9.893367587348388, −9.247615750108335, −8.727569107882461, −8.272928076366563, −7.986694282228405, −6.773085010383903, −6.557217001392604, −5.920040934986779, −5.726503781589451, −5.003381527002624, −4.259753341953308, −4.175838501170672, −3.507049987914516, −2.662795922084212, −1.927529492552475, −1.340994509576916, −0.3425349002946103,
0.3425349002946103, 1.340994509576916, 1.927529492552475, 2.662795922084212, 3.507049987914516, 4.175838501170672, 4.259753341953308, 5.003381527002624, 5.726503781589451, 5.920040934986779, 6.557217001392604, 6.773085010383903, 7.986694282228405, 8.272928076366563, 8.727569107882461, 9.247615750108335, 9.893367587348388, 10.47140372846643, 10.77397207789587, 11.39222087775162, 11.85804327172355, 12.39502350703085, 12.83513470245134, 13.34481545087183, 13.66119483827873
