L(s) = 1 | + (0.222 + 0.974i)5-s + (0.222 − 0.974i)9-s + (1.52 − 0.347i)13-s + 0.867i·17-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.24·41-s + 45-s + (0.222 − 0.974i)49-s + (0.376 + 0.781i)53-s + (1.12 + 1.40i)61-s + (0.678 + 1.40i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)5-s + (0.222 − 0.974i)9-s + (1.52 − 0.347i)13-s + 0.867i·17-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.24·41-s + 45-s + (0.222 − 0.974i)49-s + (0.376 + 0.781i)53-s + (1.12 + 1.40i)61-s + (0.678 + 1.40i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.343929545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.343929545\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 29 | \( 1 + (0.222 + 0.974i)T \) |
good | 3 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 - 0.867iT - T^{2} \) |
| 19 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 37 | \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + 1.24T + T^{2} \) |
| 43 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 83 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \) |
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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
−9.231835348721227941359901766436, −8.462360127261883241561955583625, −7.70256389641720317861911585030, −6.69725407190828192139042179187, −6.23036380824061498660215701952, −5.58240657392985194761589489921, −4.05876042741650366993923274443, −3.61028172593814311635208879132, −2.57533036828524726480071691739, −1.27911164697826269557516783261,
1.18628256905096157088563498168, 2.16739370127896279672385138360, 3.48638343347271282190708895318, 4.46174726798803484051158018184, 5.13655478460359876515496078999, 5.90481584646956947753238594347, 6.81468680516040087491065304677, 7.77423190256531777049839288021, 8.414709344273566797786192152257, 9.077221753847594640007304823488