L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 2·11-s − 12-s − 6·13-s − 14-s − 15-s + 16-s + 4·17-s + 18-s − 2·19-s + 20-s + 21-s − 2·22-s − 23-s − 24-s + 25-s − 6·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.84353020007377880364562387527, −6.80638143203647892179139124424, −6.54803922834294413878258228014, −5.56972529387403890285320862441, −4.97323881817003640205564352544, −4.49200792235238317710682715869, −3.18578848022050730727214523721, −2.62730587338856740860957759242, −1.49758147407760225858573157584, 0,
1.49758147407760225858573157584, 2.62730587338856740860957759242, 3.18578848022050730727214523721, 4.49200792235238317710682715869, 4.97323881817003640205564352544, 5.56972529387403890285320862441, 6.54803922834294413878258228014, 6.80638143203647892179139124424, 7.84353020007377880364562387527