L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s − 6·11-s + 2·12-s + 2·13-s + 4·14-s + 16-s + 18-s − 4·19-s + 8·21-s − 6·22-s + 2·24-s − 5·25-s + 2·26-s − 4·27-s + 4·28-s + 4·31-s + 32-s − 12·33-s + 36-s + 4·37-s − 4·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 1.74·21-s − 1.27·22-s + 0.408·24-s − 25-s + 0.392·26-s − 0.769·27-s + 0.755·28-s + 0.718·31-s + 0.176·32-s − 2.08·33-s + 1/6·36-s + 0.657·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.271075544\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.271075544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.88013907095950671991603611029, −9.942505353869395091050894698746, −8.551880873534616105011482646500, −8.107939322961323795970351474484, −7.46682025356819819578371193029, −5.93029628134094737560230707913, −4.99927116218852259459902565637, −4.05951678114838271873159817135, −2.77156487822388402999564657844, −1.95202314698375464079211503776,
1.95202314698375464079211503776, 2.77156487822388402999564657844, 4.05951678114838271873159817135, 4.99927116218852259459902565637, 5.93029628134094737560230707913, 7.46682025356819819578371193029, 8.107939322961323795970351474484, 8.551880873534616105011482646500, 9.942505353869395091050894698746, 10.88013907095950671991603611029