| L(s) = 1 | − 1.73·2-s − 0.684·3-s + 1.99·4-s − 5-s + 1.18·6-s − 1.28·7-s − 1.73·8-s − 0.532·9-s + 1.73·10-s − 1.87·11-s − 1.36·12-s + 2.22·14-s + 0.684·15-s + 0.999·16-s + 1.96·17-s + 0.921·18-s + 0.347·19-s − 1.99·20-s + 0.879·21-s + 3.25·22-s − 1.96·23-s + 1.18·24-s + 25-s + 1.04·27-s − 2.57·28-s − 1.18·30-s + 1.28·33-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.684·3-s + 1.99·4-s − 5-s + 1.18·6-s − 1.28·7-s − 1.73·8-s − 0.532·9-s + 1.73·10-s − 1.87·11-s − 1.36·12-s + 2.22·14-s + 0.684·15-s + 0.999·16-s + 1.96·17-s + 0.921·18-s + 0.347·19-s − 1.99·20-s + 0.879·21-s + 3.25·22-s − 1.96·23-s + 1.18·24-s + 25-s + 1.04·27-s − 2.57·28-s − 1.18·30-s + 1.28·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1055 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1055 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1411148950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1411148950\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + T^{2} \) |
| 3 | \( 1 + 0.684T + T^{2} \) |
| 7 | \( 1 + 1.28T + T^{2} \) |
| 11 | \( 1 + 1.87T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.96T + T^{2} \) |
| 19 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 + 1.96T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.684T + T^{2} \) |
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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
−10.20397902206731128713853640619, −9.443256824871072856801585350341, −8.117420079115649454794586141610, −8.015139128276044794513231751804, −7.08177012315202807751955895442, −6.08059628313512465329476644262, −5.29180717748895863419168819956, −3.49966116828873309955237286362, −2.60273844181005401864729391543, −0.53104847688163364502655595711,
0.53104847688163364502655595711, 2.60273844181005401864729391543, 3.49966116828873309955237286362, 5.29180717748895863419168819956, 6.08059628313512465329476644262, 7.08177012315202807751955895442, 8.015139128276044794513231751804, 8.117420079115649454794586141610, 9.443256824871072856801585350341, 10.20397902206731128713853640619
