| L(s) = 1 | + 3-s + 3.38·5-s + 5.02·7-s + 9-s + 1.05·11-s + 2.71·13-s + 3.38·15-s + 1.45·17-s − 7.90·19-s + 5.02·21-s − 23-s + 6.43·25-s + 27-s + 29-s − 7.96·31-s + 1.05·33-s + 16.9·35-s − 2.92·37-s + 2.71·39-s + 12.0·41-s − 9.53·43-s + 3.38·45-s − 9.69·47-s + 18.2·49-s + 1.45·51-s + 5.88·53-s + 3.55·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·5-s + 1.89·7-s + 0.333·9-s + 0.316·11-s + 0.752·13-s + 0.873·15-s + 0.352·17-s − 1.81·19-s + 1.09·21-s − 0.208·23-s + 1.28·25-s + 0.192·27-s + 0.185·29-s − 1.43·31-s + 0.182·33-s + 2.87·35-s − 0.480·37-s + 0.434·39-s + 1.88·41-s − 1.45·43-s + 0.504·45-s − 1.41·47-s + 2.60·49-s + 0.203·51-s + 0.808·53-s + 0.479·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.869890336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.869890336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.38T + 5T^{2} \) |
| 7 | \( 1 - 5.02T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 31 | \( 1 + 7.96T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 9.53T + 43T^{2} \) |
| 47 | \( 1 + 9.69T + 47T^{2} \) |
| 53 | \( 1 - 5.88T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 - 6.64T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 - 9.33T + 73T^{2} \) |
| 79 | \( 1 + 5.74T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 0.992T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.052859632089047162092791169534, −7.17694220780604675597611640280, −6.36638863882002716880141933231, −5.74545849095949121794091908884, −5.04824620184614386013764396232, −4.36810544584905262408398464465, −3.54495941054176579192519599193, −2.20843375216720360259628213109, −1.95382891200097841457085894405, −1.19140414243202936354911809775,
1.19140414243202936354911809775, 1.95382891200097841457085894405, 2.20843375216720360259628213109, 3.54495941054176579192519599193, 4.36810544584905262408398464465, 5.04824620184614386013764396232, 5.74545849095949121794091908884, 6.36638863882002716880141933231, 7.17694220780604675597611640280, 8.052859632089047162092791169534
