L(s) = 1 | + 5-s − 7-s − 2·11-s + 4·13-s + 6·17-s + 4·19-s − 23-s + 25-s − 2·29-s − 2·31-s − 35-s + 4·37-s + 2·41-s − 4·43-s − 4·47-s + 49-s + 12·53-s − 2·55-s + 6·59-s − 10·61-s + 4·65-s + 8·67-s − 4·71-s − 4·73-s + 2·77-s + 14·79-s − 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.169·35-s + 0.657·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.64·53-s − 0.269·55-s + 0.781·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s − 0.474·71-s − 0.468·73-s + 0.227·77-s + 1.57·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.045958023\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.045958023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.54658108200924, −13.24034801875910, −12.62948544119107, −12.26553811697899, −11.57305933564538, −11.26624237950501, −10.55474842390860, −10.19059793896806, −9.726458858078652, −9.275716426849164, −8.710511139682018, −8.064560874285762, −7.766452459714921, −7.099238573394531, −6.590215196055474, −5.919394271210939, −5.504806633452862, −5.247111154349935, −4.287767998916316, −3.753275758365187, −3.151587578280353, −2.745312193297038, −1.848446395918438, −1.234840996813501, −0.5739558781300467,
0.5739558781300467, 1.234840996813501, 1.848446395918438, 2.745312193297038, 3.151587578280353, 3.753275758365187, 4.287767998916316, 5.247111154349935, 5.504806633452862, 5.919394271210939, 6.590215196055474, 7.099238573394531, 7.766452459714921, 8.064560874285762, 8.710511139682018, 9.275716426849164, 9.726458858078652, 10.19059793896806, 10.55474842390860, 11.26624237950501, 11.57305933564538, 12.26553811697899, 12.62948544119107, 13.24034801875910, 13.54658108200924