L(s) = 1 | − 3-s + 9-s − 2·11-s − 6·13-s − 2·17-s + 4·23-s − 27-s − 8·31-s + 2·33-s + 2·37-s + 6·39-s − 2·41-s + 4·43-s − 8·47-s + 2·51-s + 6·53-s + 10·59-s − 2·61-s + 8·67-s − 4·69-s − 12·71-s + 4·73-s + 81-s − 4·83-s + 10·89-s + 8·93-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.485·17-s + 0.834·23-s − 0.192·27-s − 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 0.280·51-s + 0.824·53-s + 1.30·59-s − 0.256·61-s + 0.977·67-s − 0.481·69-s − 1.42·71-s + 0.468·73-s + 1/9·81-s − 0.439·83-s + 1.05·89-s + 0.829·93-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5981187925\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5981187925\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.46502048632916, −13.81281918075734, −13.06699055851846, −12.86485525121600, −12.40039445815615, −11.66929293318192, −11.39934076123971, −10.77037305243664, −10.21603178896424, −9.878003465886219, −9.165380144065562, −8.803255984035802, −7.914598343103223, −7.493416800314372, −7.021514662204049, −6.508159161516069, −5.755328623708870, −5.069033619309022, −4.995573523543875, −4.151551787589752, −3.481726484953341, −2.591950702057891, −2.227903324432214, −1.274470118903676, −0.2797460834183156,
0.2797460834183156, 1.274470118903676, 2.227903324432214, 2.591950702057891, 3.481726484953341, 4.151551787589752, 4.995573523543875, 5.069033619309022, 5.755328623708870, 6.508159161516069, 7.021514662204049, 7.493416800314372, 7.914598343103223, 8.803255984035802, 9.165380144065562, 9.878003465886219, 10.21603178896424, 10.77037305243664, 11.39934076123971, 11.66929293318192, 12.40039445815615, 12.86485525121600, 13.06699055851846, 13.81281918075734, 14.46502048632916