L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 5·11-s − 5·13-s + 15-s − 5·17-s + 21-s + 8·23-s + 25-s + 5·27-s + 29-s + 2·31-s + 5·33-s + 35-s − 4·37-s + 5·39-s + 2·41-s − 4·43-s + 2·45-s + 13·47-s + 49-s + 5·51-s + 8·53-s + 5·55-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s − 1.38·13-s + 0.258·15-s − 1.21·17-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s + 0.359·31-s + 0.870·33-s + 0.169·35-s − 0.657·37-s + 0.800·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.89·47-s + 1/7·49-s + 0.700·51-s + 1.09·53-s + 0.674·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5204105911\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5204105911\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.933090113047627386899930651835, −8.326252895426375260782489267644, −7.28341955254791884773214456096, −6.89697191740093866563607257836, −5.72298381097272128754064112273, −5.10481919271793697789070363989, −4.42069152679976638159688125579, −2.98305309112396590072777751878, −2.46327675219920633768464919345, −0.44812726391012052326964214963,
0.44812726391012052326964214963, 2.46327675219920633768464919345, 2.98305309112396590072777751878, 4.42069152679976638159688125579, 5.10481919271793697789070363989, 5.72298381097272128754064112273, 6.89697191740093866563607257836, 7.28341955254791884773214456096, 8.326252895426375260782489267644, 8.933090113047627386899930651835