| L(s) = 1 | + 2.56·3-s + (−2 − i)5-s + (−2 + 1.73i)7-s + 3.56·9-s + 0.972·11-s + 0.561i·13-s + (−5.12 − 2.56i)15-s + 4.43·17-s + 1.12i·19-s + (−5.12 + 4.43i)21-s + 1.12·23-s + (3 + 4i)25-s + 1.43·27-s + 4.43i·29-s + 8.87·31-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + (−0.894 − 0.447i)5-s + (−0.755 + 0.654i)7-s + 1.18·9-s + 0.293·11-s + 0.155i·13-s + (−1.32 − 0.661i)15-s + 1.07·17-s + 0.257i·19-s + (−1.11 + 0.968i)21-s + 0.234·23-s + (0.600 + 0.800i)25-s + 0.276·27-s + 0.823i·29-s + 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427837172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427837172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 - 0.972T + 11T^{2} \) |
| 13 | \( 1 - 0.561iT - 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 - 1.12iT - 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43iT - 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 - 8.87iT - 41T^{2} \) |
| 43 | \( 1 + 1.94iT - 43T^{2} \) |
| 47 | \( 1 + 0.972iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 7.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 8.87iT - 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−9.005596242961029936590583983120, −8.315818887123625221641000666049, −7.86082726134945052240510959510, −6.99505933086049390902576413194, −6.04924144742111610055810326253, −4.92288629092091667458877959254, −3.95400551390262022593946928463, −3.24765791756356593588590076555, −2.59264472445891814111420582619, −1.17032434772127522826850139302,
0.823727398402289446405465396944, 2.48741553227523243702193621303, 3.19833105809010581844125391243, 3.81052492138063338178588439886, 4.56327076947573676644439815707, 6.06916132772775576510316308778, 6.89749798114848939302770413759, 7.70020743275407642589449931682, 8.004005122653473056929424522809, 8.925460671979283352689061527970
