L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s + 12-s + 5·13-s − 3·14-s + 15-s + 16-s + 7·17-s − 18-s + 7·19-s + 20-s + 3·21-s − 24-s + 25-s − 5·26-s + 27-s + 3·28-s − 7·29-s − 30-s + 6·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.60·19-s + 0.223·20-s + 0.654·21-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s + 0.566·28-s − 1.29·29-s − 0.182·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.615458865\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.615458865\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.447756770228291251105623799986, −7.86708540943814559053363429939, −7.48889842910459192601994958462, −6.34964582335915713836067434463, −5.60993166683938271984093021828, −4.85944559305045742099025345093, −3.59793917567907753132968632863, −2.98399993717107095058766737944, −1.59388859600272842161552282342, −1.24510794557034026712657627305,
1.24510794557034026712657627305, 1.59388859600272842161552282342, 2.98399993717107095058766737944, 3.59793917567907753132968632863, 4.85944559305045742099025345093, 5.60993166683938271984093021828, 6.34964582335915713836067434463, 7.48889842910459192601994958462, 7.86708540943814559053363429939, 8.447756770228291251105623799986