| L(s) = 1 | + 1.91·3-s − 3.40·5-s − 0.869·7-s + 0.683·9-s − 2.80·13-s − 6.53·15-s − 7.29·17-s + 2.38·19-s − 1.66·21-s − 7.44·23-s + 6.60·25-s − 4.44·27-s + 7.34·29-s + 2.64·31-s + 2.96·35-s − 1.03·37-s − 5.38·39-s + 2.89·41-s + 2.18·43-s − 2.32·45-s + 3.94·47-s − 6.24·49-s − 13.9·51-s + 7.11·53-s + 4.57·57-s + 8.09·59-s + 2.69·61-s + ⋯ |
| L(s) = 1 | + 1.10·3-s − 1.52·5-s − 0.328·7-s + 0.227·9-s − 0.777·13-s − 1.68·15-s − 1.76·17-s + 0.546·19-s − 0.363·21-s − 1.55·23-s + 1.32·25-s − 0.855·27-s + 1.36·29-s + 0.474·31-s + 0.500·35-s − 0.169·37-s − 0.861·39-s + 0.451·41-s + 0.333·43-s − 0.346·45-s + 0.575·47-s − 0.892·49-s − 1.95·51-s + 0.977·53-s + 0.605·57-s + 1.05·59-s + 0.344·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172033235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172033235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 0.869T + 7T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 2.69T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 0.194T + 83T^{2} \) |
| 89 | \( 1 - 4.18T + 89T^{2} \) |
| 97 | \( 1 - 4.99T + 97T^{2} \) |
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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
−7.919934890089290431345206136143, −7.36008999947329897185512137965, −6.74622483801447761617597273796, −5.85540996675591587639586955535, −4.64081560877609798169194922474, −4.24243471372062521845058281792, −3.49773145246734875997783173757, −2.76131642982116377210557614307, −2.10546178935948615200050900933, −0.47851881326083029533507371912,
0.47851881326083029533507371912, 2.10546178935948615200050900933, 2.76131642982116377210557614307, 3.49773145246734875997783173757, 4.24243471372062521845058281792, 4.64081560877609798169194922474, 5.85540996675591587639586955535, 6.74622483801447761617597273796, 7.36008999947329897185512137965, 7.919934890089290431345206136143
