L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.114 − 2.23i)5-s + (−1.40 + 1.40i)7-s + (−0.707 + 0.707i)8-s + (1.49 − 1.66i)10-s + 5.29·11-s + (−1.91 + 1.91i)13-s − 1.98·14-s − 1.00·16-s + (−0.488 + 0.488i)17-s + (−2.44 + 3.60i)19-s + (2.23 − 0.114i)20-s + (3.74 + 3.74i)22-s + (3.88 + 3.88i)23-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.0512 − 0.998i)5-s + (−0.530 + 0.530i)7-s + (−0.250 + 0.250i)8-s + (0.473 − 0.524i)10-s + 1.59·11-s + (−0.532 + 0.532i)13-s − 0.530·14-s − 0.250·16-s + (−0.118 + 0.118i)17-s + (−0.561 + 0.827i)19-s + (0.499 − 0.0256i)20-s + (0.797 + 0.797i)22-s + (0.810 + 0.810i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.116309341\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.116309341\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.114 + 2.23i)T \) |
| 19 | \( 1 + (2.44 - 3.60i)T \) |
good | 7 | \( 1 + (1.40 - 1.40i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + (1.91 - 1.91i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.488 - 0.488i)T - 17iT^{2} \) |
| 23 | \( 1 + (-3.88 - 3.88i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 7.84iT - 31T^{2} \) |
| 37 | \( 1 + (-5.60 - 5.60i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.90iT - 41T^{2} \) |
| 43 | \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.86 + 7.86i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.93 - 8.93i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.611T + 59T^{2} \) |
| 61 | \( 1 - 8.31T + 61T^{2} \) |
| 67 | \( 1 + (-10.2 - 10.2i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.97iT - 71T^{2} \) |
| 73 | \( 1 + (-7.72 - 7.72i)T + 73iT^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + (5.43 + 5.43i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.42T + 89T^{2} \) |
| 97 | \( 1 + (3.10 + 3.10i)T + 97iT^{2} \) |
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
−9.330126873427479129892271007792, −8.744012960160732110989062756335, −7.931752014367061239742636487419, −6.93188582841540082257608565501, −6.18313300775280132740758124738, −5.54830769690448438425882872487, −4.37613825447472765488159191779, −3.99911818553920615604946833593, −2.62490947192484929735578389345, −1.25654858657748579906964201179,
0.77116111885728919167118354560, 2.34852994440374563240053622497, 3.18533563746248676482612595000, 4.00177678581888097414905329937, 4.83858245739599350237926404314, 6.11989156044870850419464059069, 6.75561160006386240711133793813, 7.19434700181735280265159281379, 8.534968508209237495771119030240, 9.393357524859176396181314565592