| L(s) = 1 | + 2.87·3-s + 4.22·5-s + 7-s + 5.26·9-s + 4.61·11-s − 13-s + 12.1·15-s − 7.82·17-s − 2.22·19-s + 2.87·21-s − 2.77·23-s + 12.8·25-s + 6.52·27-s − 5.74·29-s − 5.89·31-s + 13.2·33-s + 4.22·35-s + 9.18·37-s − 2.87·39-s + 3.40·41-s − 4.75·43-s + 22.2·45-s − 2.62·47-s + 49-s − 22.5·51-s − 11.8·53-s + 19.5·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 1.89·5-s + 0.377·7-s + 1.75·9-s + 1.39·11-s − 0.277·13-s + 3.13·15-s − 1.89·17-s − 0.511·19-s + 0.627·21-s − 0.578·23-s + 2.57·25-s + 1.25·27-s − 1.06·29-s − 1.05·31-s + 2.31·33-s + 0.714·35-s + 1.50·37-s − 0.460·39-s + 0.531·41-s − 0.725·43-s + 3.32·45-s − 0.382·47-s + 0.142·49-s − 3.15·51-s − 1.62·53-s + 2.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.032518013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.032518013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 17 | \( 1 + 7.82T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 9.18T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 + 4.75T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 + 2.84T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 - 8.11T + 73T^{2} \) |
| 79 | \( 1 + 8.58T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 0.944T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.935903672170805549591067986298, −8.306032452687063850588893226308, −7.23125668033818263186055937387, −6.54558630773253553685070671982, −5.87634656471150928764545450082, −4.66007841498056768830918564638, −3.97694157390528262039652150834, −2.81755332949796436347385657169, −1.98738017849708977890729334533, −1.66011703031324062654119157597,
1.66011703031324062654119157597, 1.98738017849708977890729334533, 2.81755332949796436347385657169, 3.97694157390528262039652150834, 4.66007841498056768830918564638, 5.87634656471150928764545450082, 6.54558630773253553685070671982, 7.23125668033818263186055937387, 8.306032452687063850588893226308, 8.935903672170805549591067986298
