| L(s) = 1 | + 3-s + 4.34·5-s − 1.07·7-s + 9-s − 3.41·11-s + 13-s + 4.34·15-s + 2·17-s + 1.07·19-s − 1.07·21-s + 2.15·23-s + 13.8·25-s + 27-s + 2·29-s + 5.75·31-s − 3.41·33-s − 4.68·35-s − 6.68·37-s + 39-s − 0.340·41-s + 10.8·43-s + 4.34·45-s − 7.41·47-s − 5.83·49-s + 2·51-s − 2.68·53-s − 14.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.94·5-s − 0.407·7-s + 0.333·9-s − 1.03·11-s + 0.277·13-s + 1.12·15-s + 0.485·17-s + 0.247·19-s − 0.235·21-s + 0.449·23-s + 2.76·25-s + 0.192·27-s + 0.371·29-s + 1.03·31-s − 0.595·33-s − 0.791·35-s − 1.09·37-s + 0.160·39-s − 0.0531·41-s + 1.65·43-s + 0.646·45-s − 1.08·47-s − 0.833·49-s + 0.280·51-s − 0.368·53-s − 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.732673527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.732673527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4.34T + 5T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.75T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 + 0.340T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−9.771624211229250954378068475530, −9.022670606928706011193809618885, −8.242380823516333586321271767610, −7.15848377428148763256013421456, −6.29353146137862476426063643954, −5.56345335412064371411570317002, −4.75674760138880651568274604629, −3.15597612582686089814662180394, −2.50171596612514574494959246919, −1.36957983056257743767875835517,
1.36957983056257743767875835517, 2.50171596612514574494959246919, 3.15597612582686089814662180394, 4.75674760138880651568274604629, 5.56345335412064371411570317002, 6.29353146137862476426063643954, 7.15848377428148763256013421456, 8.242380823516333586321271767610, 9.022670606928706011193809618885, 9.771624211229250954378068475530
