| L(s) = 1 | + (11.5 + 19.9i)5-s + (−9.65e3 + 4.34e4i)7-s + (−8.66e5 − 5.00e5i)11-s + 1.96e6i·13-s + (−5.15e6 + 8.92e6i)17-s + (8.66e4 − 5.00e4i)19-s + (−4.11e7 + 2.37e7i)23-s + (2.44e7 − 4.22e7i)25-s + 1.43e8i·29-s + (1.35e8 + 7.83e7i)31-s + (−9.75e5 + 3.06e5i)35-s + (−8.38e7 − 1.45e8i)37-s − 1.37e7·41-s − 1.33e9·43-s + (−7.44e8 − 1.28e9i)47-s + ⋯ |
| L(s) = 1 | + (0.00164 + 0.00285i)5-s + (−0.217 + 0.976i)7-s + (−1.62 − 0.936i)11-s + 1.46i·13-s + (−0.879 + 1.52i)17-s + (0.00802 − 0.00463i)19-s + (−1.33 + 0.768i)23-s + (0.499 − 0.866i)25-s + 1.29i·29-s + (0.850 + 0.491i)31-s + (−0.00313 + 0.000987i)35-s + (−0.198 − 0.344i)37-s − 0.0185·41-s − 1.38·43-s + (−0.473 − 0.820i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000258 + 0.999i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.000258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.06273296399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06273296399\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (9.65e3 - 4.34e4i)T \) |
| good | 5 | \( 1 + (-11.5 - 19.9i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (8.66e5 + 5.00e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.96e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (5.15e6 - 8.92e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.66e4 + 5.00e4i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (4.11e7 - 2.37e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.43e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.35e8 - 7.83e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (8.38e7 + 1.45e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.37e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.33e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (7.44e8 + 1.28e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.06e9 - 6.14e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.83e9 - 4.90e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-8.76e9 + 5.06e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (3.65e9 - 6.32e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.66e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.27e10 - 7.36e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-2.02e10 - 3.51e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 8.35e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (8.64e9 + 1.49e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.68e11iT - 7.15e21T^{2} \) |
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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
−10.79659082173066252889712735398, −9.927345373506440102365222951303, −8.614919842361494489173064880268, −8.326085179889115200564001918715, −6.77495807038930774270599869152, −5.93724576666083063771040371611, −4.98267526305276648532393471824, −3.72071257241635283133815485956, −2.54341199766073847254435223597, −1.70533986301994442210068853001,
0.01686739900205629374902081504, 0.60471314946994552928506322678, 2.23812196574614518887828179086, 3.08938811406328099582686205035, 4.48646560279092112095239942145, 5.21562917701651095657268358112, 6.55665322823176875938973493145, 7.59659636182021286068622108702, 8.115560428900499747979455333061, 9.741518187904741314410587221679
