| L(s) = 1 | − 2-s + 1.79·3-s + 4-s + 3.27·5-s − 1.79·6-s − 8-s + 0.208·9-s − 3.27·10-s + 3.66·11-s + 1.79·12-s + 3.61·13-s + 5.86·15-s + 16-s − 2.96·17-s − 0.208·18-s + 6.46·19-s + 3.27·20-s − 3.66·22-s − 3.49·23-s − 1.79·24-s + 5.72·25-s − 3.61·26-s − 5.00·27-s − 0.926·29-s − 5.86·30-s + 0.183·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.03·3-s + 0.5·4-s + 1.46·5-s − 0.731·6-s − 0.353·8-s + 0.0693·9-s − 1.03·10-s + 1.10·11-s + 0.517·12-s + 1.00·13-s + 1.51·15-s + 0.250·16-s − 0.720·17-s − 0.0490·18-s + 1.48·19-s + 0.732·20-s − 0.781·22-s − 0.727·23-s − 0.365·24-s + 1.14·25-s − 0.709·26-s − 0.962·27-s − 0.172·29-s − 1.07·30-s + 0.0328·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.390933486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.390933486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 + 0.926T + 29T^{2} \) |
| 31 | \( 1 - 0.183T + 31T^{2} \) |
| 37 | \( 1 - 2.94T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 + 9.98T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 + 6.62T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 - 5.61T + 73T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.051325514393561659131949196841, −7.29659593488962685378848922931, −6.41979787993785480328544767167, −6.03724104121907447153906552311, −5.26912783417946177849591433613, −4.03813866181900009929048838389, −3.32054193112376165748959963967, −2.46714021304146877307277089566, −1.80525969605597330850257419435, −1.04281087473596258873921868094,
1.04281087473596258873921868094, 1.80525969605597330850257419435, 2.46714021304146877307277089566, 3.32054193112376165748959963967, 4.03813866181900009929048838389, 5.26912783417946177849591433613, 6.03724104121907447153906552311, 6.41979787993785480328544767167, 7.29659593488962685378848922931, 8.051325514393561659131949196841
