| L(s) = 1 | + 1.08·3-s − 4.19·7-s − 1.81·9-s − 6.43·11-s − 2.24·13-s + 7.84·17-s − 19-s − 4.57·21-s + 0.859·23-s − 5.24·27-s + 8.38·29-s + 1.24·31-s − 7.00·33-s + 6.79·37-s − 2.44·39-s + 5.92·41-s − 6.81·43-s + 6.00·47-s + 10.6·49-s + 8.55·51-s − 13.7·53-s − 1.08·57-s + 6.88·59-s − 1.31·61-s + 7.60·63-s − 4.73·67-s + 0.936·69-s + ⋯ |
| L(s) = 1 | + 0.629·3-s − 1.58·7-s − 0.603·9-s − 1.93·11-s − 0.622·13-s + 1.90·17-s − 0.229·19-s − 0.998·21-s + 0.179·23-s − 1.00·27-s + 1.55·29-s + 0.223·31-s − 1.22·33-s + 1.11·37-s − 0.392·39-s + 0.925·41-s − 1.03·43-s + 0.876·47-s + 1.51·49-s + 1.19·51-s − 1.88·53-s − 0.144·57-s + 0.896·59-s − 0.168·61-s + 0.958·63-s − 0.578·67-s + 0.112·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230594208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230594208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 + 6.43T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 7.84T + 17T^{2} \) |
| 23 | \( 1 - 0.859T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 6.88T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 9.86T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 5.42T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−8.260527061686267909731442160855, −7.950136948845861959628957833212, −7.16556119509388174887259787848, −6.19313416139401745224242988340, −5.58741050448752143219939993653, −4.79735791811941317414249959678, −3.49013750588492898404309316889, −2.91657404336742598428573155388, −2.47561651512472468204297282998, −0.58249917694669068976413874887,
0.58249917694669068976413874887, 2.47561651512472468204297282998, 2.91657404336742598428573155388, 3.49013750588492898404309316889, 4.79735791811941317414249959678, 5.58741050448752143219939993653, 6.19313416139401745224242988340, 7.16556119509388174887259787848, 7.950136948845861959628957833212, 8.260527061686267909731442160855
