L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s + 12-s + 2·13-s − 14-s + 2·15-s + 16-s + 2·17-s − 18-s + 2·20-s + 21-s − 24-s − 25-s − 2·26-s + 27-s + 28-s + 2·29-s − 2·30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.439107936\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.439107936\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.222342273474573779535958878238, −7.76606514900733948109427526908, −6.87204263221520597540152628060, −6.15764371174594687030190015062, −5.49907117448076233936059018391, −4.51889757866883921400584450355, −3.52521205154929257820001588956, −2.61508625734212454439516390514, −1.84572904708575779361666937440, −0.973410034959001153441111592272,
0.973410034959001153441111592272, 1.84572904708575779361666937440, 2.61508625734212454439516390514, 3.52521205154929257820001588956, 4.51889757866883921400584450355, 5.49907117448076233936059018391, 6.15764371174594687030190015062, 6.87204263221520597540152628060, 7.76606514900733948109427526908, 8.222342273474573779535958878238