L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s − 0.999·27-s + (−0.499 − 0.866i)28-s + 1.73i·31-s − 0.999·36-s − 37-s + (1.5 − 0.866i)39-s + 1.73i·43-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s − 0.999·27-s + (−0.499 − 0.866i)28-s + 1.73i·31-s − 0.999·36-s − 37-s + (1.5 − 0.866i)39-s + 1.73i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.776730985\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776730985\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - 1.73iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−8.737746830402323761778979237602, −7.971484709414468762173790282772, −7.12322501217349445116362600144, −6.60572401394442785051130352317, −5.98000098094200671993789129484, −4.92486245114663513451730362960, −3.91341765843496044833779666399, −2.94763947132601989060785544738, −1.64491124363510222152541210383, −1.23657015470512178342727386420,
1.91908160474091000860488401732, 2.78150680868566652829432988207, 3.59762279093051355611247593360, 4.24079695779039540127202302746, 5.44634112571062042917450517953, 5.92538701598272688267222747163, 7.11804143952243880207139070366, 7.978756701008942807421859585060, 8.558061667358138619338182477038, 8.852073757962957248510621342561