L(s) = 1 | + 1.61·3-s + 1.61i·5-s − 7-s + 1.61·9-s − 13-s + 2.61i·15-s + 0.618i·17-s − 1.61·21-s + i·23-s − 1.61·25-s + 27-s + i·29-s + i·31-s − 1.61i·35-s + 37-s + ⋯ |
L(s) = 1 | + 1.61·3-s + 1.61i·5-s − 7-s + 1.61·9-s − 13-s + 2.61i·15-s + 0.618i·17-s − 1.61·21-s + i·23-s − 1.61·25-s + 27-s + i·29-s + i·31-s − 1.61i·35-s + 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.831995102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831995102\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 887 | \( 1 - iT \) |
good | 3 | \( 1 - 1.61T + T^{2} \) |
| 5 | \( 1 - 1.61iT - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - 0.618iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + 0.618iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 0.618T + T^{2} \) |
| 53 | \( 1 + 1.61iT - T^{2} \) |
| 59 | \( 1 - 1.61T + T^{2} \) |
| 61 | \( 1 + 1.61iT - T^{2} \) |
| 67 | \( 1 + 0.618iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 - 1.61T + T^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 - 0.618iT - T^{2} \) |
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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
−9.007364641944372449548444299583, −8.092466386780539129131689528658, −7.43868062135450680217608902291, −6.89877038653869053408694955255, −6.30576286197199453648603526509, −5.09283184770782619551100747284, −3.59256396459331022322017541903, −3.52366273635151565081048246760, −2.63882900145015680334175332499, −1.99179484128894203214207238228,
0.843352674290698245485302930228, 2.29822591145127699637954461721, 2.78236982181001584987069518016, 4.01717972452714273540709413058, 4.43123957398982467478147383586, 5.42090385221704261061073648018, 6.40560789873799127751681638124, 7.44373775355514557032438432208, 7.930148168811535263174497837490, 8.688390737408038544918600524001