| L(s) = 1 | + 3-s + 3.46i·5-s + 9-s + 3.46i·11-s − 1.73i·13-s + 3.46i·15-s − 5·19-s + 6.92i·23-s − 6.99·25-s + 27-s + 5·31-s + 3.46i·33-s − 11·37-s − 1.73i·39-s − 3.46i·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54i·5-s + 0.333·9-s + 1.04i·11-s − 0.480i·13-s + 0.894i·15-s − 1.14·19-s + 1.44i·23-s − 1.39·25-s + 0.192·27-s + 0.898·31-s + 0.603i·33-s − 1.80·37-s − 0.277i·39-s − 0.541i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642417924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642417924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.66iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 8.66iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 5.19iT - 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.409863477005114765813697953850, −8.406825439633831400881201267445, −7.62622813778368307672790550775, −7.03054961388024306225890718685, −6.43282934919576666577401721914, −5.39929973568992063165783240205, −4.26276336308682154602837989785, −3.43425875021722192515375802399, −2.63509226060092959808889592459, −1.78546889058288277760838198860,
0.49293210020831007669943317233, 1.65609198074412514788105944751, 2.75765159939395673957920250623, 4.00913351942227560404143369319, 4.52389517164719368798028541267, 5.47021868472603074523878989202, 6.29677724126691030585079720232, 7.24366023348999556481093641612, 8.346557778780206596675586920066, 8.724419783202658386347118158521
