| L(s) = 1 | + 3.29·3-s − 3.40·5-s + 1.01·7-s + 7.85·9-s + 5.72·11-s + 2.46·13-s − 11.2·15-s + 5.23·17-s − 4.81·19-s + 3.32·21-s − 5.25·23-s + 6.58·25-s + 15.9·27-s + 5.28·29-s + 8.08·31-s + 18.8·33-s − 3.43·35-s − 4.26·37-s + 8.12·39-s − 12.5·41-s + 3.79·43-s − 26.7·45-s + 47-s − 5.97·49-s + 17.2·51-s + 4.27·53-s − 19.4·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 1.52·5-s + 0.381·7-s + 2.61·9-s + 1.72·11-s + 0.683·13-s − 2.89·15-s + 1.26·17-s − 1.10·19-s + 0.726·21-s − 1.09·23-s + 1.31·25-s + 3.07·27-s + 0.981·29-s + 1.45·31-s + 3.28·33-s − 0.581·35-s − 0.701·37-s + 1.30·39-s − 1.96·41-s + 0.579·43-s − 3.98·45-s + 0.145·47-s − 0.854·49-s + 2.41·51-s + 0.587·53-s − 2.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.114617371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.114617371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 - 8.08T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 - 5.15T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.30T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 9.84T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.290481940047955661846164268101, −7.66686148474426439698714533068, −6.88852651532549482160231754133, −6.29572361113010790011630289521, −4.67118169639314699881863398469, −4.13344147789491032310855459231, −3.61697526981452785214872203928, −3.09588486056027330545027213599, −1.86870016021856458755457719802, −1.07085967110242905492063537871,
1.07085967110242905492063537871, 1.86870016021856458755457719802, 3.09588486056027330545027213599, 3.61697526981452785214872203928, 4.13344147789491032310855459231, 4.67118169639314699881863398469, 6.29572361113010790011630289521, 6.88852651532549482160231754133, 7.66686148474426439698714533068, 8.290481940047955661846164268101
