| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 3·9-s − 2·10-s + 4·11-s + 4·13-s + 14-s + 16-s + 8·17-s + 3·18-s + 2·19-s + 2·20-s − 4·22-s − 25-s − 4·26-s − 28-s + 2·29-s − 6·31-s − 32-s − 8·34-s − 2·35-s − 3·36-s + 10·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.707·18-s + 0.458·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.371·29-s − 1.07·31-s − 0.176·32-s − 1.37·34-s − 0.338·35-s − 1/2·36-s + 1.64·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.027638188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027638188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.88344761387658947503598101916, −7.37784559335912029840696264493, −6.28664941237184942144060567566, −5.92214710684541833965840069767, −5.56528813677650326141936298733, −4.15496858462953392394573093342, −3.34561366065603947604442305947, −2.67929126530051147823213321473, −1.52536654790130646687896240029, −0.877236400762477963039132426390,
0.877236400762477963039132426390, 1.52536654790130646687896240029, 2.67929126530051147823213321473, 3.34561366065603947604442305947, 4.15496858462953392394573093342, 5.56528813677650326141936298733, 5.92214710684541833965840069767, 6.28664941237184942144060567566, 7.37784559335912029840696264493, 7.88344761387658947503598101916
