L(s) = 1 | − 3-s + 2·5-s − 7-s − 2·9-s + 2·11-s + 13-s − 2·15-s + 6·19-s + 21-s + 23-s − 25-s + 5·27-s − 29-s + 31-s − 2·33-s − 2·35-s + 6·37-s − 39-s + 3·41-s − 4·45-s − 3·47-s + 49-s − 6·53-s + 4·55-s − 6·57-s − 8·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 1.37·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.962·27-s − 0.185·29-s + 0.179·31-s − 0.348·33-s − 0.338·35-s + 0.986·37-s − 0.160·39-s + 0.468·41-s − 0.596·45-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.794·57-s − 1.04·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.903292499\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.903292499\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.69739487498136, −16.15704279491517, −15.59639873274711, −14.73643583308358, −14.26679006056994, −13.69235640578993, −13.28007437622685, −12.45362554639928, −11.98057996678066, −11.25638166689724, −10.96310484217827, −10.00339464753138, −9.567203873547474, −9.119572201960376, −8.284880253179866, −7.586881461773473, −6.685480038247013, −6.253162100921862, −5.618110340764693, −5.167858244117217, −4.207133840651601, −3.299703141058737, −2.636217999855838, −1.590640948134176, −0.6926816306774573,
0.6926816306774573, 1.590640948134176, 2.636217999855838, 3.299703141058737, 4.207133840651601, 5.167858244117217, 5.618110340764693, 6.253162100921862, 6.685480038247013, 7.586881461773473, 8.284880253179866, 9.119572201960376, 9.567203873547474, 10.00339464753138, 10.96310484217827, 11.25638166689724, 11.98057996678066, 12.45362554639928, 13.28007437622685, 13.69235640578993, 14.26679006056994, 14.73643583308358, 15.59639873274711, 16.15704279491517, 16.69739487498136