| L(s) = 1 | − 3-s − 1.82·5-s − 1.82·7-s − 2·9-s − 0.828·11-s − 13-s + 1.82·15-s − 4.65·17-s − 4·19-s + 1.82·21-s − 8.82·23-s − 1.65·25-s + 5·27-s + 8.82·29-s + 0.828·33-s + 3.34·35-s − 1.82·37-s + 39-s − 2.82·41-s − 3·43-s + 3.65·45-s + 5.48·47-s − 3.65·49-s + 4.65·51-s + 8.82·53-s + 1.51·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.817·5-s − 0.691·7-s − 0.666·9-s − 0.249·11-s − 0.277·13-s + 0.472·15-s − 1.12·17-s − 0.917·19-s + 0.398·21-s − 1.84·23-s − 0.331·25-s + 0.962·27-s + 1.63·29-s + 0.144·33-s + 0.565·35-s − 0.300·37-s + 0.160·39-s − 0.441·41-s − 0.457·43-s + 0.545·45-s + 0.800·47-s − 0.522·49-s + 0.652·51-s + 1.21·53-s + 0.204·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3008542409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3008542409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 3T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−8.453648969556990158510835829242, −8.048894399465562319912232631491, −6.98126162895045127333724890153, −6.36398245623373141613671556369, −5.75729702678906529671710018159, −4.67600692732073850707254285948, −4.07809289046269523549995192374, −3.06350024827991364431747594783, −2.13516872854691533778213401245, −0.31284490625544040699614037646,
0.31284490625544040699614037646, 2.13516872854691533778213401245, 3.06350024827991364431747594783, 4.07809289046269523549995192374, 4.67600692732073850707254285948, 5.75729702678906529671710018159, 6.36398245623373141613671556369, 6.98126162895045127333724890153, 8.048894399465562319912232631491, 8.453648969556990158510835829242
