L(s) = 1 | − 0.571i·2-s + 1.67·4-s + 1.42i·7-s − 2.10i·8-s − 2.67·11-s − 4.67i·13-s + 0.816·14-s + 2.14·16-s − 2.67i·17-s − 4.67·19-s + 1.52i·22-s − 5.91i·23-s − 2.67·26-s + 2.38i·28-s − 9.48·29-s + ⋯ |
L(s) = 1 | − 0.404i·2-s + 0.836·4-s + 0.539i·7-s − 0.742i·8-s − 0.805·11-s − 1.29i·13-s + 0.218·14-s + 0.535·16-s − 0.648i·17-s − 1.07·19-s + 0.325i·22-s − 1.23i·23-s − 0.524·26-s + 0.451i·28-s − 1.76·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.631435742\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.631435742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.571iT - 2T^{2} \) |
| 7 | \( 1 - 1.42iT - 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 + 4.67iT - 13T^{2} \) |
| 17 | \( 1 + 2.67iT - 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 5.91iT - 23T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 + 1.81iT - 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 0.471iT - 43T^{2} \) |
| 47 | \( 1 + 6.95iT - 47T^{2} \) |
| 53 | \( 1 + 1.14iT - 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 6.59iT - 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.71iT - 73T^{2} \) |
| 79 | \( 1 - 0.287T + 79T^{2} \) |
| 83 | \( 1 + 4.28iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 7.83iT - 97T^{2} \) |
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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
−8.861916105551927712415262624228, −8.064586667101783794370653816409, −7.41345637283224495911065697865, −6.46715735319880807156873339821, −5.75276071607080040603002376121, −4.93019418729748588230826160393, −3.68876819046779209407770845528, −2.69883611767869445878467107357, −2.16862004576120117999274931428, −0.52937148981819215720915322442,
1.56948993312864499089330514179, 2.43822992707068209071111139658, 3.65597158131288093758237146006, 4.54538333236910961869503309748, 5.62753600444219418844563247217, 6.31018499657270860339075457178, 7.08748932018200558308625121595, 7.69817611065262648701202143868, 8.418444331502381519513020987984, 9.377495058744093577401045368598