| L(s) = 1 | − 3·5-s − 7-s − 5·11-s − 3·17-s + 19-s − 23-s + 4·25-s + 5·29-s + 8·31-s + 3·35-s − 37-s − 12·41-s + 11·43-s + 8·47-s + 49-s − 10·53-s + 15·55-s − 8·59-s + 13·61-s − 2·67-s + 8·71-s − 11·73-s + 5·77-s + 2·79-s − 18·83-s + 9·85-s + 18·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 1.50·11-s − 0.727·17-s + 0.229·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s + 1.43·31-s + 0.507·35-s − 0.164·37-s − 1.87·41-s + 1.67·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 2.02·55-s − 1.04·59-s + 1.66·61-s − 0.244·67-s + 0.949·71-s − 1.28·73-s + 0.569·77-s + 0.225·79-s − 1.97·83-s + 0.976·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5771985778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5771985778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.96748051554612, −13.33212937267633, −12.92826149390892, −12.42903765011118, −11.87049071718093, −11.61270951009594, −10.87372643562501, −10.52296023889303, −10.06697490738826, −9.451139298306180, −8.628701433186057, −8.430721660240331, −7.764780482282466, −7.495714064684986, −6.826179946684389, −6.312312419397757, −5.633552947593689, −4.922893684851817, −4.551026705358869, −3.950209115297345, −3.263330904978313, −2.765938855232092, −2.206713842940749, −1.065455383121086, −0.2767305000043675,
0.2767305000043675, 1.065455383121086, 2.206713842940749, 2.765938855232092, 3.263330904978313, 3.950209115297345, 4.551026705358869, 4.922893684851817, 5.633552947593689, 6.312312419397757, 6.826179946684389, 7.495714064684986, 7.764780482282466, 8.430721660240331, 8.628701433186057, 9.451139298306180, 10.06697490738826, 10.52296023889303, 10.87372643562501, 11.61270951009594, 11.87049071718093, 12.42903765011118, 12.92826149390892, 13.33212937267633, 13.96748051554612
