| L(s) = 1 | + 1.81·3-s − 2.66·5-s − 7-s + 0.295·9-s + 4.22·11-s − 1.10·13-s − 4.84·15-s + 5.14·17-s − 1.59·19-s − 1.81·21-s − 4.83·23-s + 2.12·25-s − 4.90·27-s + 4.03·29-s + 2.30·31-s + 7.66·33-s + 2.66·35-s − 0.485·37-s − 2.00·39-s + 4.14·41-s + 4.91·43-s − 0.788·45-s + 6.97·47-s + 49-s + 9.33·51-s + 6.45·53-s − 11.2·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s − 1.19·5-s − 0.377·7-s + 0.0984·9-s + 1.27·11-s − 0.306·13-s − 1.25·15-s + 1.24·17-s − 0.364·19-s − 0.396·21-s − 1.00·23-s + 0.425·25-s − 0.944·27-s + 0.749·29-s + 0.414·31-s + 1.33·33-s + 0.451·35-s − 0.0798·37-s − 0.321·39-s + 0.646·41-s + 0.748·43-s − 0.117·45-s + 1.01·47-s + 0.142·49-s + 1.30·51-s + 0.887·53-s − 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.062725021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062725021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 5.14T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 0.485T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 - 6.45T + 53T^{2} \) |
| 59 | \( 1 - 0.825T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.98T + 67T^{2} \) |
| 71 | \( 1 + 7.69T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.548196840202008161712422405489, −7.82751914271248431853124778753, −7.37915327379630398686969136524, −6.44006094789872771634383805437, −5.62098617268344150609114989894, −4.30188617141328410424488259873, −3.84137116037968899066261654721, −3.16883105908673963333134888995, −2.20941018279175403782102109496, −0.800591627639453611884882072453,
0.800591627639453611884882072453, 2.20941018279175403782102109496, 3.16883105908673963333134888995, 3.84137116037968899066261654721, 4.30188617141328410424488259873, 5.62098617268344150609114989894, 6.44006094789872771634383805437, 7.37915327379630398686969136524, 7.82751914271248431853124778753, 8.548196840202008161712422405489
