| L(s) = 1 | − 3·3-s − 20·7-s + 9·9-s + 16·11-s − 58·13-s − 38·17-s + 4·19-s + 60·21-s + 80·23-s − 27·27-s + 82·29-s − 8·31-s − 48·33-s − 426·37-s + 174·39-s − 246·41-s + 524·43-s + 464·47-s + 57·49-s + 114·51-s + 702·53-s − 12·57-s − 592·59-s + 574·61-s − 180·63-s + 172·67-s − 240·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.07·7-s + 1/3·9-s + 0.438·11-s − 1.23·13-s − 0.542·17-s + 0.0482·19-s + 0.623·21-s + 0.725·23-s − 0.192·27-s + 0.525·29-s − 0.0463·31-s − 0.253·33-s − 1.89·37-s + 0.714·39-s − 0.937·41-s + 1.85·43-s + 1.44·47-s + 0.166·49-s + 0.313·51-s + 1.81·53-s − 0.0278·57-s − 1.30·59-s + 1.20·61-s − 0.359·63-s + 0.313·67-s − 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.042813389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042813389\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 426 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 702 T + p^{3} T^{2} \) |
| 59 | \( 1 + 592 T + p^{3} T^{2} \) |
| 61 | \( 1 - 574 T + p^{3} T^{2} \) |
| 67 | \( 1 - 172 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 558 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 164 T + p^{3} T^{2} \) |
| 89 | \( 1 + 510 T + p^{3} T^{2} \) |
| 97 | \( 1 + 514 T + p^{3} T^{2} \) |
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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
−10.23965090362842956723123245373, −9.500284390924219199191105834868, −8.690171446706379576113595608279, −7.23080795725470797881076881103, −6.78108586531915716653875275574, −5.71686697259282903780373595666, −4.74648194001397939375542189649, −3.58810115669747616246960065486, −2.33830843650467447170750058822, −0.60587640455480600067627560368,
0.60587640455480600067627560368, 2.33830843650467447170750058822, 3.58810115669747616246960065486, 4.74648194001397939375542189649, 5.71686697259282903780373595666, 6.78108586531915716653875275574, 7.23080795725470797881076881103, 8.690171446706379576113595608279, 9.500284390924219199191105834868, 10.23965090362842956723123245373
