| L(s) = 1 | + 1.33·2-s + 0.790·4-s − 0.279·8-s − 1.16·16-s + 1.82·17-s + 1.33·19-s − 0.209·23-s + 1.82·31-s − 1.27·32-s + 2.44·34-s + 1.79·38-s − 0.279·46-s + 0.618·47-s + 49-s − 1.95·53-s − 1.95·61-s + 2.44·62-s − 0.547·64-s + 1.44·68-s + 1.05·76-s − 1.95·79-s − 1.95·83-s − 0.165·92-s + 0.827·94-s + 1.33·98-s − 2.61·106-s − 1.61·107-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.790·4-s − 0.279·8-s − 1.16·16-s + 1.82·17-s + 1.33·19-s − 0.209·23-s + 1.82·31-s − 1.27·32-s + 2.44·34-s + 1.79·38-s − 0.279·46-s + 0.618·47-s + 49-s − 1.95·53-s − 1.95·61-s + 2.44·62-s − 0.547·64-s + 1.44·68-s + 1.05·76-s − 1.95·79-s − 1.95·83-s − 0.165·92-s + 0.827·94-s + 1.33·98-s − 2.61·106-s − 1.61·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.538436045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538436045\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.33T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.82T + T^{2} \) |
| 19 | \( 1 - 1.33T + T^{2} \) |
| 23 | \( 1 + 0.209T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.82T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.618T + T^{2} \) |
| 53 | \( 1 + 1.95T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.95T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.95T + T^{2} \) |
| 83 | \( 1 + 1.95T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.757559404778757287260412450887, −7.84074538217986916015680000977, −7.23053151582233900614204327311, −6.18778841810694993695568679747, −5.69712574898164366499901375615, −4.96009851236727662115523264904, −4.22696648210132150076729840492, −3.25941969567978494669220389188, −2.81707069808496930006828393459, −1.28555287507302653788962362903,
1.28555287507302653788962362903, 2.81707069808496930006828393459, 3.25941969567978494669220389188, 4.22696648210132150076729840492, 4.96009851236727662115523264904, 5.69712574898164366499901375615, 6.18778841810694993695568679747, 7.23053151582233900614204327311, 7.84074538217986916015680000977, 8.757559404778757287260412450887
