L(s) = 1 | + (−0.541 − 0.541i)2-s − 0.414i·4-s + (−0.382 − 0.923i)5-s + (−0.765 + 0.765i)8-s + (−0.292 + 0.707i)10-s − 0.765i·11-s + (−0.707 − 0.707i)13-s + 0.414·16-s + (−0.382 + 0.158i)20-s + (−0.414 + 0.414i)22-s + (−0.707 + 0.707i)25-s + 0.765i·26-s + (0.541 + 0.541i)32-s + (1 + 0.414i)40-s − 1.84i·41-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.541i)2-s − 0.414i·4-s + (−0.382 − 0.923i)5-s + (−0.765 + 0.765i)8-s + (−0.292 + 0.707i)10-s − 0.765i·11-s + (−0.707 − 0.707i)13-s + 0.414·16-s + (−0.382 + 0.158i)20-s + (−0.414 + 0.414i)22-s + (−0.707 + 0.707i)25-s + 0.765i·26-s + (0.541 + 0.541i)32-s + (1 + 0.414i)40-s − 1.84i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5434603687\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5434603687\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 0.765iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.84T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.84iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.47491852799362796014399304232, −9.762751858319427314569905978455, −8.799854915718864510827157696324, −8.318435452824471949653106141041, −7.13968050604446613271019429800, −5.69818673999793289560039417269, −5.16152028384679902468942590721, −3.75146405964570847438051736670, −2.32369453557618542888148133738, −0.76263624706000351682124792638,
2.41460360310530724210722720962, 3.59008916874264094497740478474, 4.66997268944348490895000365137, 6.26655755857492201074577143831, 6.99829100007556552456065230010, 7.59264461117258332536905574925, 8.467868407815004936532586641882, 9.545179143707910087542589284762, 10.11881645754644177956993619880, 11.34553975704963622605779311378