| L(s) = 1 | − 1.30·3-s + 5-s − 7-s − 1.30·9-s + 2.30·11-s + 3.60·13-s − 1.30·15-s + 1.69·17-s − 19-s + 1.30·21-s − 0.394·23-s − 4·25-s + 5.60·27-s + 4.30·29-s − 8.30·31-s − 3·33-s − 35-s − 3.60·37-s − 4.69·39-s + 0.302·41-s − 7.21·43-s − 1.30·45-s − 7.60·47-s + 49-s − 2.21·51-s + 3.90·53-s + 2.30·55-s + ⋯ |
| L(s) = 1 | − 0.752·3-s + 0.447·5-s − 0.377·7-s − 0.434·9-s + 0.694·11-s + 1.00·13-s − 0.336·15-s + 0.411·17-s − 0.229·19-s + 0.284·21-s − 0.0822·23-s − 0.800·25-s + 1.07·27-s + 0.799·29-s − 1.49·31-s − 0.522·33-s − 0.169·35-s − 0.592·37-s − 0.752·39-s + 0.0472·41-s − 1.09·43-s − 0.194·45-s − 1.10·47-s + 0.142·49-s − 0.309·51-s + 0.536·53-s + 0.310·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 23 | \( 1 + 0.394T + 23T^{2} \) |
| 29 | \( 1 - 4.30T + 29T^{2} \) |
| 31 | \( 1 + 8.30T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 0.302T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 5.60T + 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.29749828403317015060283646172, −6.42078469128637526946448960770, −6.19724295916709496656896228886, −5.47957057244520050929038464506, −4.81576371580389174728787659441, −3.75883354917440937422857922078, −3.25662256996197630860446733836, −2.06496155617604062130041094237, −1.18644667425728392720254936323, 0,
1.18644667425728392720254936323, 2.06496155617604062130041094237, 3.25662256996197630860446733836, 3.75883354917440937422857922078, 4.81576371580389174728787659441, 5.47957057244520050929038464506, 6.19724295916709496656896228886, 6.42078469128637526946448960770, 7.29749828403317015060283646172
