| L(s) = 1 | − 10·5-s + 8·7-s + 40·11-s + 13·13-s − 130·17-s + 20·19-s − 25·25-s + 18·29-s + 184·31-s − 80·35-s − 74·37-s + 362·41-s − 76·43-s − 452·47-s − 279·49-s − 382·53-s − 400·55-s + 464·59-s + 358·61-s − 130·65-s + 700·67-s − 748·71-s + 1.05e3·73-s + 320·77-s + 976·79-s − 1.00e3·83-s + 1.30e3·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.431·7-s + 1.09·11-s + 0.277·13-s − 1.85·17-s + 0.241·19-s − 1/5·25-s + 0.115·29-s + 1.06·31-s − 0.386·35-s − 0.328·37-s + 1.37·41-s − 0.269·43-s − 1.40·47-s − 0.813·49-s − 0.990·53-s − 0.980·55-s + 1.02·59-s + 0.751·61-s − 0.248·65-s + 1.27·67-s − 1.25·71-s + 1.69·73-s + 0.473·77-s + 1.38·79-s − 1.33·83-s + 1.65·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 17 | \( 1 + 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 452 T + p^{3} T^{2} \) |
| 53 | \( 1 + 382 T + p^{3} T^{2} \) |
| 59 | \( 1 - 464 T + p^{3} T^{2} \) |
| 61 | \( 1 - 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 748 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1058 T + p^{3} T^{2} \) |
| 79 | \( 1 - 976 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 - 386 T + p^{3} T^{2} \) |
| 97 | \( 1 + 614 T + p^{3} T^{2} \) |
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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
−8.415835799532833144252639398581, −7.83880587546268550898239253240, −6.77566341692342541737202464532, −6.35750713434191452402935545161, −5.02547568463668778879419598596, −4.27004576808736719714988191151, −3.63523474890028917746978469684, −2.37005685936039322077740699884, −1.22381567310171597205156908387, 0,
1.22381567310171597205156908387, 2.37005685936039322077740699884, 3.63523474890028917746978469684, 4.27004576808736719714988191151, 5.02547568463668778879419598596, 6.35750713434191452402935545161, 6.77566341692342541737202464532, 7.83880587546268550898239253240, 8.415835799532833144252639398581
