| L(s) = 1 | − 0.890·2-s − 3-s − 1.20·4-s − 2.94·5-s + 0.890·6-s + 7-s + 2.85·8-s + 9-s + 2.62·10-s + 0.179·11-s + 1.20·12-s − 1.10·13-s − 0.890·14-s + 2.94·15-s − 0.128·16-s + 5.29·17-s − 0.890·18-s + 7.81·19-s + 3.55·20-s − 21-s − 0.159·22-s − 0.108·23-s − 2.85·24-s + 3.67·25-s + 0.984·26-s − 27-s − 1.20·28-s + ⋯ |
| L(s) = 1 | − 0.629·2-s − 0.577·3-s − 0.603·4-s − 1.31·5-s + 0.363·6-s + 0.377·7-s + 1.00·8-s + 0.333·9-s + 0.829·10-s + 0.0541·11-s + 0.348·12-s − 0.306·13-s − 0.237·14-s + 0.760·15-s − 0.0321·16-s + 1.28·17-s − 0.209·18-s + 1.79·19-s + 0.795·20-s − 0.218·21-s − 0.0340·22-s − 0.0226·23-s − 0.582·24-s + 0.735·25-s + 0.192·26-s − 0.192·27-s − 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9855454759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9855454759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.890T + 2T^{2} \) |
| 5 | \( 1 + 2.94T + 5T^{2} \) |
| 11 | \( 1 - 0.179T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 + 0.108T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 6.33T + 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−7.87735150076813537649814272258, −7.48321082161164505204542118317, −6.64050534031483975357011365204, −5.57560753544359954663435264812, −4.97813006924827110775438705602, −4.35157388029554068134776923740, −3.67177419304119246669850341668, −2.75316101075224671249083332052, −1.07688591919141700502708661107, −0.75322040111704202158453596170,
0.75322040111704202158453596170, 1.07688591919141700502708661107, 2.75316101075224671249083332052, 3.67177419304119246669850341668, 4.35157388029554068134776923740, 4.97813006924827110775438705602, 5.57560753544359954663435264812, 6.64050534031483975357011365204, 7.48321082161164505204542118317, 7.87735150076813537649814272258
