L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.777 + 0.974i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.277 + 1.21i)10-s + (1.62 + 0.781i)13-s + (−0.900 − 0.433i)16-s − 0.445·17-s + (−0.900 − 0.433i)18-s + (−1.12 + 0.541i)20-s + (−0.123 + 0.541i)25-s + (0.400 + 1.75i)26-s + (−0.222 − 0.974i)32-s + (−0.277 − 0.347i)34-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.777 + 0.974i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.277 + 1.21i)10-s + (1.62 + 0.781i)13-s + (−0.900 − 0.433i)16-s − 0.445·17-s + (−0.900 − 0.433i)18-s + (−1.12 + 0.541i)20-s + (−0.123 + 0.541i)25-s + (0.400 + 1.75i)26-s + (−0.222 − 0.974i)32-s + (−0.277 − 0.347i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.802396689\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802396689\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 5 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 37 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
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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
−8.840209615014542794145175365638, −8.355213640672442485291148297593, −7.46223937006311593900704047127, −6.53639303189906664411549592518, −6.23592781709660231809453958417, −5.57053688753933778692280291558, −4.59523089867574250956592843294, −3.64260597683414204613203554132, −2.87492795964932702809552206742, −1.96867285347024772883967750196,
0.886895070456441456411712866099, 1.80030725281333404655264044284, 2.96894162155472768342237359063, 3.65478700234436113141541644284, 4.70986265014394613309595709112, 5.37893740987856483132596425278, 6.07073334827788455864257500275, 6.49391804007849976914136195841, 8.104579232618447923082322506364, 8.689399374803312428283901153333