| L(s) = 1 | + i·5-s + 1.42i·7-s − 3.99·11-s − 13-s + 5.18i·17-s + 2.01i·19-s + 4.23·23-s − 25-s − 0.499i·29-s + 7.99i·31-s − 1.42·35-s − 11.7·37-s − 1.64i·41-s + 2.01i·43-s + 9.06·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.537i·7-s − 1.20·11-s − 0.277·13-s + 1.25i·17-s + 0.461i·19-s + 0.882·23-s − 0.200·25-s − 0.0927i·29-s + 1.43i·31-s − 0.240·35-s − 1.92·37-s − 0.257i·41-s + 0.306i·43-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3813317472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3813317472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 1.42iT - 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 17 | \( 1 - 5.18iT - 17T^{2} \) |
| 19 | \( 1 - 2.01iT - 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 0.499iT - 29T^{2} \) |
| 31 | \( 1 - 7.99iT - 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 1.64iT - 41T^{2} \) |
| 43 | \( 1 - 2.01iT - 43T^{2} \) |
| 47 | \( 1 - 9.06T + 47T^{2} \) |
| 53 | \( 1 - 0.889iT - 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 6.48iT - 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 7.07iT - 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 2.64iT - 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.135701929032830099600011292272, −7.40814445255444966833255431044, −6.81090794214535734547433086117, −6.00183578750429785232219190582, −5.39424380520787452951256624382, −4.81076053440673967230279300133, −3.76497819588985804482317585140, −3.07904274849554400191272913260, −2.33528440483130401308728278221, −1.46954396867773569721614964075,
0.095293553468838043588200761651, 0.943585614859513862832679866580, 2.21909439820302606951299017298, 2.86818498259443780540966557873, 3.77990235912068878532464894856, 4.69322645509580706182585290764, 5.12699500354882205242298791532, 5.77199187688619069285612306593, 6.81447827693206406409447542965, 7.36338954201867188779948821810
