| L(s) = 1 | + 3.03·3-s − 5-s + 0.865·7-s + 6.20·9-s + 4.43·13-s − 3.03·15-s + 0.550·17-s − 0.258·19-s + 2.62·21-s − 7.92·23-s + 25-s + 9.72·27-s + 9.36·29-s + 5.31·31-s − 0.865·35-s + 6.38·37-s + 13.4·39-s − 8.74·41-s + 6.38·43-s − 6.20·45-s − 0.823·47-s − 6.25·49-s + 1.66·51-s + 2.40·53-s − 0.785·57-s + 13.5·59-s + 6.60·61-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 0.447·5-s + 0.327·7-s + 2.06·9-s + 1.23·13-s − 0.783·15-s + 0.133·17-s − 0.0594·19-s + 0.573·21-s − 1.65·23-s + 0.200·25-s + 1.87·27-s + 1.73·29-s + 0.954·31-s − 0.146·35-s + 1.05·37-s + 2.15·39-s − 1.36·41-s + 0.973·43-s − 0.925·45-s − 0.120·47-s − 0.892·49-s + 0.233·51-s + 0.329·53-s − 0.104·57-s + 1.75·59-s + 0.846·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.602362055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.602362055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.03T + 3T^{2} \) |
| 7 | \( 1 - 0.865T + 7T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 - 0.550T + 17T^{2} \) |
| 19 | \( 1 + 0.258T + 19T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 - 6.38T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 + 0.823T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 + 2.79T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−8.040575866929151887925657629431, −7.16971903722968853987946457781, −6.51700892176005385552127148558, −5.69250686579455724076687189628, −4.45799777333773899925219027847, −4.14703510188529024540166211230, −3.34469115981564830805821302263, −2.72176603642704429402501772591, −1.87400418147478726143578182248, −0.990929318509037426035827373345,
0.990929318509037426035827373345, 1.87400418147478726143578182248, 2.72176603642704429402501772591, 3.34469115981564830805821302263, 4.14703510188529024540166211230, 4.45799777333773899925219027847, 5.69250686579455724076687189628, 6.51700892176005385552127148558, 7.16971903722968853987946457781, 8.040575866929151887925657629431
