L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.541 − 0.541i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s − 0.414i·9-s − 1.41i·11-s + 0.765i·12-s + (−0.541 + 0.541i)13-s + i·16-s + (0.382 + 0.158i)18-s − 19-s + (1.30 + 0.541i)22-s + (−0.707 − 0.292i)24-s + (−0.292 − 0.707i)26-s + (−0.765 + 0.765i)27-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.541 − 0.541i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s − 0.414i·9-s − 1.41i·11-s + 0.765i·12-s + (−0.541 + 0.541i)13-s + i·16-s + (0.382 + 0.158i)18-s − 19-s + (1.30 + 0.541i)22-s + (−0.707 − 0.292i)24-s + (−0.292 − 0.707i)26-s + (−0.765 + 0.765i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3877761227\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3877761227\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
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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
−8.971705605492976818468406186685, −8.411297325572516574176361139304, −7.46077752104511353046476501859, −6.76263559246322000920699573525, −6.11506495281474142309961161283, −5.52964384902852039967394271784, −4.47674126739951430638698534366, −3.42856697223432367901445884125, −1.75290790538994615801277609397, −0.34300803069080927988521601884,
1.73855866296181391514161069727, 2.64282515902634475345637575612, 3.87814318723081949484059291431, 4.75696439551653255228287740965, 5.13748970487831157689641867481, 6.51382471474814745550431487502, 7.51392407277433178510816857005, 8.147101955261457568737478215988, 9.096519976602568315936798928236, 9.986678742571154276236645095291