| L(s) = 1 | − i·2-s − 4-s + 5-s + 0.891i·7-s + i·8-s − i·10-s + 0.713·11-s − 4.24·13-s + 0.891·14-s + 16-s + 0.522·17-s − 2.05i·19-s − 20-s − 0.713i·22-s + (−1.20 − 4.64i)23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.337i·7-s + 0.353i·8-s − 0.316i·10-s + 0.214·11-s − 1.17·13-s + 0.238·14-s + 0.250·16-s + 0.126·17-s − 0.472i·19-s − 0.223·20-s − 0.152i·22-s + (−0.251 − 0.967i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.461842064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461842064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (1.20 + 4.64i)T \) |
| good | 7 | \( 1 - 0.891iT - 7T^{2} \) |
| 11 | \( 1 - 0.713T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 0.522T + 17T^{2} \) |
| 19 | \( 1 + 2.05iT - 19T^{2} \) |
| 29 | \( 1 + 3.14iT - 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 - 4.32iT - 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 - 8.11iT - 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.77iT - 59T^{2} \) |
| 61 | \( 1 + 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 0.835iT - 67T^{2} \) |
| 71 | \( 1 - 4.15iT - 71T^{2} \) |
| 73 | \( 1 - 4.80T + 73T^{2} \) |
| 79 | \( 1 + 1.14iT - 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 - 9.10T + 89T^{2} \) |
| 97 | \( 1 - 4.69iT - 97T^{2} \) |
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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
−9.096905567708490342837467540276, −8.283085656218937715375615937217, −7.37146276540635901988395454822, −6.47132348952403288300501173348, −5.54585392898016310198533057193, −4.77926281016227415204067744657, −3.90585071546692231287113516211, −2.65027912824302139404340402146, −2.12471113582238875926675544135, −0.56442456790118267468628352210,
1.20932110791408262184393672017, 2.58099990397584892951241560374, 3.72689911318784957490701393560, 4.70835688659607482173428799921, 5.41512620926806882998835148166, 6.26545523863871338523628216823, 7.05403681595689018405494572643, 7.67619892098869959421407212714, 8.507530756510602475428626217107, 9.338498161231691444373806341089
