L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s − 4·11-s + 2·13-s − 4·14-s + 16-s + 20-s + 4·22-s + 2·23-s + 25-s − 2·26-s + 4·28-s + 2·29-s − 31-s − 32-s + 4·35-s + 2·37-s − 40-s + 4·43-s − 4·44-s − 2·46-s + 9·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.223·20-s + 0.852·22-s + 0.417·23-s + 1/5·25-s − 0.392·26-s + 0.755·28-s + 0.371·29-s − 0.179·31-s − 0.176·32-s + 0.676·35-s + 0.328·37-s − 0.158·40-s + 0.609·43-s − 0.603·44-s − 0.294·46-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.699687406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699687406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.641350139907664306787752683725, −8.150796596553767652422868563846, −7.52992246495075111010282900919, −6.68270021911821628143898093985, −5.59762019130118498212445002711, −5.14276933439147365736148700871, −4.10451103708232674588558534526, −2.76122532106114812239317963033, −1.97400009635083133161984619011, −0.936445013975751983415827368279,
0.936445013975751983415827368279, 1.97400009635083133161984619011, 2.76122532106114812239317963033, 4.10451103708232674588558534526, 5.14276933439147365736148700871, 5.59762019130118498212445002711, 6.68270021911821628143898093985, 7.52992246495075111010282900919, 8.150796596553767652422868563846, 8.641350139907664306787752683725