| L(s) = 1 | + 3.24·3-s + 1.62·5-s − 7-s + 7.50·9-s − 4.09·11-s + 1.35·13-s + 5.26·15-s − 17-s + 2.21·19-s − 3.24·21-s + 9.44·23-s − 2.36·25-s + 14.6·27-s − 4.61·29-s + 8.18·31-s − 13.2·33-s − 1.62·35-s + 6.03·37-s + 4.38·39-s + 2.10·41-s − 11.2·43-s + 12.1·45-s + 13.3·47-s + 49-s − 3.24·51-s − 3.20·53-s − 6.65·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.726·5-s − 0.377·7-s + 2.50·9-s − 1.23·11-s + 0.375·13-s + 1.35·15-s − 0.242·17-s + 0.507·19-s − 0.707·21-s + 1.97·23-s − 0.472·25-s + 2.81·27-s − 0.856·29-s + 1.47·31-s − 2.31·33-s − 0.274·35-s + 0.992·37-s + 0.702·39-s + 0.328·41-s − 1.71·43-s + 1.81·45-s + 1.95·47-s + 0.142·49-s − 0.453·51-s − 0.440·53-s − 0.896·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.382676880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.382676880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 9.44T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 6.03T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 3.20T + 53T^{2} \) |
| 59 | \( 1 + 9.76T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.22T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 9.74T + 97T^{2} \) |
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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
−8.501554620352705014459742151661, −7.88475433422427403010220261381, −7.25672372562850693314573977427, −6.46751260259917361698255619895, −5.41919965975727148061013309806, −4.58230140505840740072620066707, −3.56868947277915367643466219661, −2.82412179562553741933908870071, −2.34206228897425742150960122620, −1.20481512937232326646169834092,
1.20481512937232326646169834092, 2.34206228897425742150960122620, 2.82412179562553741933908870071, 3.56868947277915367643466219661, 4.58230140505840740072620066707, 5.41919965975727148061013309806, 6.46751260259917361698255619895, 7.25672372562850693314573977427, 7.88475433422427403010220261381, 8.501554620352705014459742151661
