L(s) = 1 | − 1.41·3-s − 0.585·5-s − 7-s − 0.999·9-s + 2·11-s − 2.82·13-s + 0.828·15-s + 2.58·17-s + 2·19-s + 1.41·21-s − 23-s − 4.65·25-s + 5.65·27-s − 8.48·29-s + 3.41·31-s − 2.82·33-s + 0.585·35-s + 10.4·37-s + 4.00·39-s + 6·41-s + 8·43-s + 0.585·45-s + 4.58·47-s + 49-s − 3.65·51-s − 0.828·53-s − 1.17·55-s + ⋯ |
L(s) = 1 | − 0.816·3-s − 0.261·5-s − 0.377·7-s − 0.333·9-s + 0.603·11-s − 0.784·13-s + 0.213·15-s + 0.627·17-s + 0.458·19-s + 0.308·21-s − 0.208·23-s − 0.931·25-s + 1.08·27-s − 1.57·29-s + 0.613·31-s − 0.492·33-s + 0.0990·35-s + 1.72·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s + 0.0873·45-s + 0.668·47-s + 0.142·49-s − 0.512·51-s − 0.113·53-s − 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9323894904\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9323894904\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 + 0.828T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 - 0.343T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−9.668662998044469234612794740431, −9.037157055887227910559764720833, −7.82832061828698379765962161220, −7.26459131032776754036660057359, −6.08894602803365857136099280618, −5.70163354976129799102519445746, −4.58302357892637675132837474401, −3.63138512705989748431462557109, −2.42882037742026536884183445571, −0.72646998985229436529144416923,
0.72646998985229436529144416923, 2.42882037742026536884183445571, 3.63138512705989748431462557109, 4.58302357892637675132837474401, 5.70163354976129799102519445746, 6.08894602803365857136099280618, 7.26459131032776754036660057359, 7.82832061828698379765962161220, 9.037157055887227910559764720833, 9.668662998044469234612794740431