L(s) = 1 | + i·2-s + i·3-s − 4-s + (−1.30 + 1.81i)5-s − 6-s − 4.63i·7-s − i·8-s − 9-s + (−1.81 − 1.30i)10-s + 5.23·11-s − i·12-s − 3.39i·13-s + 4.63·14-s + (−1.81 − 1.30i)15-s + 16-s + 2.43i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.581 + 0.813i)5-s − 0.408·6-s − 1.75i·7-s − 0.353i·8-s − 0.333·9-s + (−0.575 − 0.411i)10-s + 1.57·11-s − 0.288i·12-s − 0.942i·13-s + 1.23·14-s + (−0.469 − 0.336i)15-s + 0.250·16-s + 0.589i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28889 + 0.662543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28889 + 0.662543i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 4.63iT - 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 3.39iT - 13T^{2} \) |
| 17 | \( 1 - 2.43iT - 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 23 | \( 1 - 3.43iT - 23T^{2} \) |
| 29 | \( 1 + 0.794T + 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 - 4.63iT - 43T^{2} \) |
| 47 | \( 1 + 1.56iT - 47T^{2} \) |
| 53 | \( 1 + 9.43iT - 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 + 7.46iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 5.36T + 89T^{2} \) |
| 97 | \( 1 - 0.191iT - 97T^{2} \) |
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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
−9.996885988258591135292505658330, −9.596999695201920799730399477067, −8.266356920546430463523799036228, −7.53514940191952649280334868246, −6.93012852839441842776122818060, −6.05988814232510381737744993669, −4.80462145462906018445902877862, −3.71518288851732628121666787720, −3.51210471927325030614819529792, −0.933250612832069360313266429736,
1.10873804122669826591092970611, 2.21988188673473671521923161394, 3.42449187422810904860881637858, 4.57518180656865574116441096227, 5.48022236056030972948458749566, 6.43212986623542424732250820957, 7.54958052116997577577626274223, 8.607430707853634076756304858731, 9.166625709314877646044984456094, 9.500197507261322378282750446867