| L(s) = 1 | + 3-s + 4.41·5-s − 7-s + 9-s + 1.67·11-s − 4.66·13-s + 4.41·15-s + 6.24·17-s + 0.694·19-s − 21-s + 23-s + 14.4·25-s + 27-s + 5.60·29-s − 4.24·31-s + 1.67·33-s − 4.41·35-s + 9.26·37-s − 4.66·39-s − 5.15·41-s − 4.20·43-s + 4.41·45-s + 1.92·47-s + 49-s + 6.24·51-s − 1.84·53-s + 7.39·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.97·5-s − 0.377·7-s + 0.333·9-s + 0.505·11-s − 1.29·13-s + 1.14·15-s + 1.51·17-s + 0.159·19-s − 0.218·21-s + 0.208·23-s + 2.89·25-s + 0.192·27-s + 1.03·29-s − 0.763·31-s + 0.291·33-s − 0.746·35-s + 1.52·37-s − 0.747·39-s − 0.805·41-s − 0.641·43-s + 0.658·45-s + 0.280·47-s + 0.142·49-s + 0.875·51-s − 0.252·53-s + 0.997·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.098831192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.098831192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 0.694T + 19T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 0.581T + 79T^{2} \) |
| 83 | \( 1 + 6.95T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.88537866236376362427499295290, −7.00670996341514390530284143839, −6.55867533639454494011630912821, −5.68293154991321636868551624141, −5.25671111263647621753952227508, −4.39351093953093536585607861061, −3.18050328037875543113774941949, −2.70009586894463871053497662549, −1.85614530091720609714460312718, −1.04720123295918110502710195734,
1.04720123295918110502710195734, 1.85614530091720609714460312718, 2.70009586894463871053497662549, 3.18050328037875543113774941949, 4.39351093953093536585607861061, 5.25671111263647621753952227508, 5.68293154991321636868551624141, 6.55867533639454494011630912821, 7.00670996341514390530284143839, 7.88537866236376362427499295290
