L(s) = 1 | + (1.32 − 2.5i)2-s + (−4.5 − 6.61i)4-s + 18.5i·7-s + (−22.4 + 2.5i)8-s + 68i·11-s + (46.3 + 24.5i)14-s + (−23.5 + 59.5i)16-s + (170 + 89.9i)22-s + 40i·23-s + 125·25-s + (122. − 83.3i)28-s − 264.·29-s + (117. + 137.5i)32-s + 450·37-s + 534. i·43-s + (449. − 306i)44-s + ⋯ |
L(s) = 1 | + (0.467 − 0.883i)2-s + (−0.562 − 0.826i)4-s + 0.999i·7-s + (−0.993 + 0.110i)8-s + 1.86i·11-s + (0.883 + 0.467i)14-s + (−0.367 + 0.930i)16-s + (1.64 + 0.871i)22-s + 0.362i·23-s + 25-s + (0.826 − 0.562i)28-s − 1.69·29-s + (0.650 + 0.759i)32-s + 1.99·37-s + 1.89i·43-s + (1.54 − 1.04i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.572153438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.572153438\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 + 2.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 18.5iT \) |
good | 5 | \( 1 - 125T^{2} \) |
| 11 | \( 1 - 68iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 40iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 450T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 - 534. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 809. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 688iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 238. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−11.75998132456248402113874283640, −10.94716116799399683187046198055, −9.640172256020815748257323819809, −9.326284136725414133692564237249, −7.83572666372642928320224660389, −6.42824318617436594264038875435, −5.24712642532636152294908578754, −4.33606879044897551499129064947, −2.79028682231560371880666751428, −1.69413507839960358821850902578,
0.54076605476600933213483944613, 3.16944898960241999465414319284, 4.15596408366096363760701874839, 5.48432212707174744456581854766, 6.42211041390783834181816801503, 7.47387344450084288929845412540, 8.367220658054699017534499144671, 9.308484269733225457540696933778, 10.71669783548167037475246212983, 11.48357462008547870825451856010