L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 6·11-s − 4·13-s + 15-s − 6·17-s − 19-s + 2·21-s + 25-s − 27-s − 8·31-s + 6·33-s + 2·35-s + 8·37-s + 4·39-s − 12·41-s − 2·43-s − 45-s − 3·49-s + 6·51-s − 6·53-s + 6·55-s + 57-s + 12·59-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.43·31-s + 1.04·33-s + 0.338·35-s + 1.31·37-s + 0.640·39-s − 1.87·41-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.809·55-s + 0.132·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1881004708\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1881004708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.249280881369556848595379389670, −7.47486546641950266603937112595, −6.93929346133186867217820325138, −6.18838298240809326845294415159, −5.21462215917711114886086280445, −4.82756803415784004984035070623, −3.82794301538735459183423175699, −2.82027419172399898745573141712, −2.09085356046109669711702219006, −0.22863414088953228094422158125,
0.22863414088953228094422158125, 2.09085356046109669711702219006, 2.82027419172399898745573141712, 3.82794301538735459183423175699, 4.82756803415784004984035070623, 5.21462215917711114886086280445, 6.18838298240809326845294415159, 6.93929346133186867217820325138, 7.47486546641950266603937112595, 8.249280881369556848595379389670