| L(s) = 1 | − 1.16i·5-s + 4.00i·7-s − 1.34·11-s + 2.84·13-s + 4.41i·17-s − 7.02i·19-s − 4.58·23-s + 3.64·25-s + 2.59i·29-s + 5.66i·31-s + 4.67·35-s + 37-s + 9.38i·41-s − 9.95i·43-s − 12.2·47-s + ⋯ |
| L(s) = 1 | − 0.521i·5-s + 1.51i·7-s − 0.404·11-s + 0.789·13-s + 1.07i·17-s − 1.61i·19-s − 0.955·23-s + 0.728·25-s + 0.482i·29-s + 1.01i·31-s + 0.790·35-s + 0.164·37-s + 1.46i·41-s − 1.51i·43-s − 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298208354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298208354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 1.16iT - 5T^{2} \) |
| 7 | \( 1 - 4.00iT - 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 - 4.41iT - 17T^{2} \) |
| 19 | \( 1 + 7.02iT - 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 2.59iT - 29T^{2} \) |
| 31 | \( 1 - 5.66iT - 31T^{2} \) |
| 41 | \( 1 - 9.38iT - 41T^{2} \) |
| 43 | \( 1 + 9.95iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 9.53iT - 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 1.48iT - 67T^{2} \) |
| 71 | \( 1 - 0.505T + 71T^{2} \) |
| 73 | \( 1 - 1.93T + 73T^{2} \) |
| 79 | \( 1 + 4.99iT - 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 - 3.64iT - 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−8.593487052249372099946260291632, −7.970439840636787159246362505120, −6.78790001501962753171442128437, −6.27705323395552047815147452364, −5.37131028477400361959288699659, −5.01878735568290274875641652188, −3.97814971158777719590495671735, −2.99493457718256668502743646528, −2.24147228407536452897347687891, −1.21968176915030013835214878791,
0.35733278311690605274179552173, 1.46646738369845126808632857959, 2.59645687294321342918443049353, 3.67135075483132316075752918103, 3.98380391834725752948238768346, 5.01945956428139976937173999471, 5.89026548756992125140863184567, 6.65043001385664629267245426258, 7.20781575897972690684521225143, 7.999770925684282026590832918594
