L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 2·13-s + 15-s − 2·19-s + 2·21-s + 25-s − 27-s − 8·31-s + 33-s + 2·35-s + 2·37-s − 2·39-s − 2·43-s − 45-s − 3·49-s + 6·53-s + 55-s + 2·57-s + 12·59-s + 2·61-s − 2·63-s − 2·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.43·31-s + 0.174·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-s + 0.256·61-s − 0.251·63-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9448419645\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9448419645\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.867210729448237508015837384646, −8.086271054754877428311825479240, −7.22147591026612199052006405671, −6.56659423896616238366347042412, −5.81230406564188744733335412920, −5.02240081234243941184036798059, −4.00139660166883866031995858001, −3.33471856791961230902138906908, −2.08096164230858474674230917660, −0.61190912173332380675685594750,
0.61190912173332380675685594750, 2.08096164230858474674230917660, 3.33471856791961230902138906908, 4.00139660166883866031995858001, 5.02240081234243941184036798059, 5.81230406564188744733335412920, 6.56659423896616238366347042412, 7.22147591026612199052006405671, 8.086271054754877428311825479240, 8.867210729448237508015837384646