L(s) = 1 | + 5-s + 2·7-s − 11-s + 2·13-s + 2·19-s + 25-s + 8·31-s + 2·35-s + 2·37-s + 2·43-s − 3·49-s − 6·53-s − 55-s + 12·59-s + 2·61-s + 2·65-s − 4·67-s + 2·73-s − 2·77-s − 10·79-s + 12·83-s + 6·89-s + 4·91-s + 2·95-s + 14·97-s − 4·103-s + 12·107-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s + 0.554·13-s + 0.458·19-s + 1/5·25-s + 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.134·55-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.234·73-s − 0.227·77-s − 1.12·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.205·95-s + 1.42·97-s − 0.394·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.182019050\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.182019050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−9.147231330768035948133860610786, −8.326350192735332185448447875749, −7.75976166708726338515121473137, −6.75888529317405913099706507891, −5.96324102832559157582134241037, −5.13900536434453166716756839866, −4.37696360732250492487057417989, −3.22634087889258268749054020892, −2.17437433404389747311113154017, −1.05479763369531237820875298266,
1.05479763369531237820875298266, 2.17437433404389747311113154017, 3.22634087889258268749054020892, 4.37696360732250492487057417989, 5.13900536434453166716756839866, 5.96324102832559157582134241037, 6.75888529317405913099706507891, 7.75976166708726338515121473137, 8.326350192735332185448447875749, 9.147231330768035948133860610786