L(s) = 1 | + (0.980 − 0.195i)2-s + (−0.275 + 0.275i)3-s + (0.923 − 0.382i)4-s + (0.707 + 0.707i)5-s + (−0.216 + 0.324i)6-s + (0.831 − 0.555i)8-s + 0.847i·9-s + (0.831 + 0.555i)10-s + (−0.541 − 0.541i)11-s + (−0.149 + 0.360i)12-s + (−0.785 + 0.785i)13-s − 0.390·15-s + (0.707 − 0.707i)16-s + (0.165 + 0.831i)18-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)2-s + (−0.275 + 0.275i)3-s + (0.923 − 0.382i)4-s + (0.707 + 0.707i)5-s + (−0.216 + 0.324i)6-s + (0.831 − 0.555i)8-s + 0.847i·9-s + (0.831 + 0.555i)10-s + (−0.541 − 0.541i)11-s + (−0.149 + 0.360i)12-s + (−0.785 + 0.785i)13-s − 0.390·15-s + (0.707 − 0.707i)16-s + (0.165 + 0.831i)18-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.008927095\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.008927095\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.980 + 0.195i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.275 - 0.275i)T - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 13 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.38 + 1.38i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.275 - 0.275i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 67 | \( 1 + (-1.17 + 1.17i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.11T + T^{2} \) |
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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
−9.980898609034925871481866146042, −9.184581342558964764529100573844, −7.79220331053933186008375766753, −7.09369912196107114594190565535, −6.32026836784592651645885555537, −5.31186614944519150826678738200, −4.99552160372563130453683415572, −3.73145915166867665954956467448, −2.68964057463263771071671511687, −1.95152654687008861253748121843,
1.45109227449260233901855680106, 2.64805818661978646105171407314, 3.69384653816155534446816457643, 4.92170380458628994513871382092, 5.36022747119414466577737072606, 6.19429103737996887725231760960, 7.00549110503852222152297293355, 7.81689515699738097677295201959, 8.682077171539676615344000924874, 9.911336799129460737649372225040