L(s) = 1 | + (1.37 − 1.05i)3-s + (−0.403 + 2.29i)5-s + (2.60 + 0.435i)7-s + (0.762 − 2.90i)9-s + (1.79 − 0.315i)11-s + (−2.58 + 3.07i)13-s + (1.86 + 3.56i)15-s + 1.25·17-s + 5.76i·19-s + (4.04 − 2.16i)21-s + (5.50 − 6.55i)23-s + (−0.386 − 0.140i)25-s + (−2.02 − 4.78i)27-s + (−4.11 − 4.90i)29-s + (2.63 + 7.24i)31-s + ⋯ |
L(s) = 1 | + (0.791 − 0.610i)3-s + (−0.180 + 1.02i)5-s + (0.986 + 0.164i)7-s + (0.254 − 0.967i)9-s + (0.540 − 0.0952i)11-s + (−0.715 + 0.853i)13-s + (0.482 + 0.921i)15-s + 0.304·17-s + 1.32i·19-s + (0.881 − 0.471i)21-s + (1.14 − 1.36i)23-s + (−0.0772 − 0.0281i)25-s + (−0.389 − 0.921i)27-s + (−0.763 − 0.909i)29-s + (0.473 + 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17321 + 0.180489i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17321 + 0.180489i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 + (-2.60 - 0.435i)T \) |
good | 5 | \( 1 + (0.403 - 2.29i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.79 + 0.315i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.58 - 3.07i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 - 5.76iT - 19T^{2} \) |
| 23 | \( 1 + (-5.50 + 6.55i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.11 + 4.90i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.63 - 7.24i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 3.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.98 + 1.66i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.77 + 1.73i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-5.64 - 2.05i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.91 - 1.68i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.51 + 6.30i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.74 + 10.2i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.64 + 9.31i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.63 - 2.67i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.35 + 4.24i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.709 - 4.02i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.74 - 6.49i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 - 6.53T + 89T^{2} \) |
| 97 | \( 1 + (-3.30 + 9.08i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.37355632235761949699538946348, −9.404365749849942333457164610475, −8.515840361488759648624627218034, −7.77942015151458666556826830740, −6.95731842944756037671432114218, −6.31110820007558771744448059920, −4.82044924091812094629450644266, −3.68771356635128811012034244328, −2.63096476183216877262751516865, −1.56508703770219932223642119649,
1.24647308894434605008719075368, 2.71239996518504083092787017511, 3.98067062256672830627434999207, 4.88649577375936077218835747220, 5.38126094102885512392803916926, 7.26984668209240404288826285337, 7.80092980419356000170179118441, 8.827976907505879928275983320339, 9.201683310366092702973897825399, 10.19385299944550930831539089404