L(s) = 1 | + (−2.40 + 2.40i)5-s − 11.7·7-s + (34.7 + 34.7i)11-s + (3.17 − 3.17i)13-s − 98.0i·17-s + (15.9 + 15.9i)19-s − 69.6i·23-s + 113. i·25-s + (15.9 + 15.9i)29-s − 121. i·31-s + (28.2 − 28.2i)35-s + (37.0 + 37.0i)37-s + 59.3·41-s + (241. − 241. i)43-s − 395.·47-s + ⋯ |
L(s) = 1 | + (−0.215 + 0.215i)5-s − 0.632·7-s + (0.952 + 0.952i)11-s + (0.0678 − 0.0678i)13-s − 1.39i·17-s + (0.192 + 0.192i)19-s − 0.631i·23-s + 0.907i·25-s + (0.102 + 0.102i)29-s − 0.702i·31-s + (0.136 − 0.136i)35-s + (0.164 + 0.164i)37-s + 0.225·41-s + (0.857 − 0.857i)43-s − 1.22·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.666606192\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666606192\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.40 - 2.40i)T - 125iT^{2} \) |
| 7 | \( 1 + 11.7T + 343T^{2} \) |
| 11 | \( 1 + (-34.7 - 34.7i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-3.17 + 3.17i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 98.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-15.9 - 15.9i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 69.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-15.9 - 15.9i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 121. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-37.0 - 37.0i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 59.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-241. + 241. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 395.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (458. - 458. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-257. - 257. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-373. + 373. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-648. - 648. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 787. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 382. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-491. + 491. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 624.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−9.549201201555274758328866055125, −8.952522536494690889773242951279, −7.74571526297211461969266425019, −7.03399205042372878540605585880, −6.39468699118576013269020720733, −5.25564917404373904014126345223, −4.29102099188236775321396485536, −3.36183800590334729831379966763, −2.30125449753163202496841436708, −0.901357983599850634782591295731,
0.51354832171429263656689888064, 1.71355378133012100231113666368, 3.22745387070012195461913377233, 3.85008134389582688288455797456, 4.96390612773082137297579002139, 6.24499076615070473390843691941, 6.43608968638593690850657789493, 7.79836902686493757384865804877, 8.481981003032991446273402736305, 9.259559468132732558673882384221