L(s) = 1 | + i·2-s − 1.43i·3-s − 4-s + 1.43·6-s + 3.08i·7-s − i·8-s + 0.950·9-s − 6.46·11-s + 1.43i·12-s − 3.95i·13-s − 3.08·14-s + 16-s − 3.43i·17-s + 0.950i·18-s − 3.08·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.826i·3-s − 0.5·4-s + 0.584·6-s + 1.16i·7-s − 0.353i·8-s + 0.316·9-s − 1.95·11-s + 0.413i·12-s − 1.09i·13-s − 0.825·14-s + 0.250·16-s − 0.832i·17-s + 0.224i·18-s − 0.708·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5245661030\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5245661030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 1.43iT - 3T^{2} \) |
| 7 | \( 1 - 3.08iT - 7T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 13 | \( 1 + 3.95iT - 13T^{2} \) |
| 17 | \( 1 + 3.43iT - 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 29 | \( 1 + 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 + 7.03iT - 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 3.90iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 + 5.90iT - 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.03iT - 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−9.300457561933635429849668799000, −8.512724543714406314448906827277, −7.54778049521821499404863843087, −7.44508412799672235713284993068, −5.98693114029952959313923734529, −5.57778050711897890423261307286, −4.66043827352412813309012107324, −3.01182051812716714310107235149, −2.12572910393576819568297270070, −0.21796593097137714526795225984,
1.64778265089839964617259187778, 2.96239023006359401949048792511, 4.11973678190586253740737030984, 4.49262621326057561544454997090, 5.53436210961633479786582690392, 6.84158478020766761801411552888, 7.72725600707832388562184896869, 8.553374421385525067653688698620, 9.625876225215222728654414908039, 10.15949267786694263820499857127