| L(s) = 1 | − 0.414·3-s + 3.82·5-s + 1.58·7-s − 2.82·9-s + 4.82·11-s + 13-s − 1.58·15-s + 17-s + 5.65·19-s − 0.656·21-s + 3.17·23-s + 9.65·25-s + 2.41·27-s − 7.65·29-s − 7.65·31-s − 1.99·33-s + 6.07·35-s + 7·37-s − 0.414·39-s − 1.65·41-s − 5.58·43-s − 10.8·45-s + 9.24·47-s − 4.48·49-s − 0.414·51-s − 7.65·53-s + 18.4·55-s + ⋯ |
| L(s) = 1 | − 0.239·3-s + 1.71·5-s + 0.599·7-s − 0.942·9-s + 1.45·11-s + 0.277·13-s − 0.409·15-s + 0.242·17-s + 1.29·19-s − 0.143·21-s + 0.661·23-s + 1.93·25-s + 0.464·27-s − 1.42·29-s − 1.37·31-s − 0.348·33-s + 1.02·35-s + 1.15·37-s − 0.0663·39-s − 0.258·41-s − 0.851·43-s − 1.61·45-s + 1.34·47-s − 0.640·49-s − 0.0580·51-s − 1.05·53-s + 2.49·55-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
| \(L(1)\) |
\(\approx\) |
\(2.856468117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.856468117\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.934543643962284056926957656445, −7.921664020698377126032800888726, −6.93177295068258131016676560408, −6.28217349679446390635509805133, −5.45832383980899916810603348883, −5.26649038529834374795553042955, −3.90488981867575797611773491787, −2.93660116098150582636381792105, −1.85718173409751564615874947928, −1.14122990556000968641477029369,
1.14122990556000968641477029369, 1.85718173409751564615874947928, 2.93660116098150582636381792105, 3.90488981867575797611773491787, 5.26649038529834374795553042955, 5.45832383980899916810603348883, 6.28217349679446390635509805133, 6.93177295068258131016676560408, 7.921664020698377126032800888726, 8.934543643962284056926957656445
