| L(s) = 1 | − 2.64i·3-s + 5-s − 2·7-s − 4.00·9-s − 2.64i·11-s + 5·13-s − 2.64i·15-s + 5.29i·17-s + 5.29i·19-s + 5.29i·21-s + 6·23-s − 4·25-s + 2.64i·27-s + (−1 − 5.29i)29-s + 2.64i·31-s + ⋯ |
| L(s) = 1 | − 1.52i·3-s + 0.447·5-s − 0.755·7-s − 1.33·9-s − 0.797i·11-s + 1.38·13-s − 0.683i·15-s + 1.28i·17-s + 1.21i·19-s + 1.15i·21-s + 1.25·23-s − 0.800·25-s + 0.509i·27-s + (−0.185 − 0.982i)29-s + 0.475i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.817181 - 0.677212i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817181 - 0.677212i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + (1 + 5.29i)T \) |
| good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 5.29iT - 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 31 | \( 1 - 2.64iT - 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 + 5.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.93iT - 43T^{2} \) |
| 47 | \( 1 + 7.93iT - 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 5.29iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 2.64iT - 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 5.29iT - 89T^{2} \) |
| 97 | \( 1 - 5.29iT - 97T^{2} \) |
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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
−13.34924684784480913552501031541, −12.57156156737379614916407131246, −11.46404793409669965356109809442, −10.25475190170627145434171778233, −8.742874493889395478290563940354, −7.87529861398798680728963547354, −6.35570806238188015824100744160, −6.02071832667706757065760761607, −3.39739732224506275679364929632, −1.50679528450003839441860314788,
3.08808685831935640145183700853, 4.43739594559277891528076962187, 5.61983243082601145682414998739, 7.06487408227525129793152854033, 9.090378683942619914044714318202, 9.399564312813529369362579074415, 10.56442536522991048073397422435, 11.34773848136740402885803148359, 12.90690889584280208748276950296, 13.82837841297672472574810458594
