L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s − 13-s − 14-s − 15-s + 16-s + 3·17-s − 18-s + 4·19-s − 20-s + 21-s − 24-s + 25-s + 26-s + 27-s + 28-s + 3·29-s + 30-s + 2·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.182·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.88765591028297, −14.38023180260625, −13.92687585849549, −13.40499189104834, −12.69063972957545, −12.09759596470051, −11.79173033279350, −11.24747605780093, −10.55442872545323, −10.02931778858304, −9.717431221622909, −8.989618197944799, −8.479051173307921, −8.034183777540630, −7.635246996145451, −6.885923774771646, −6.675732739253659, −5.524989889937297, −5.225887948371785, −4.377894676749654, −3.690261940469079, −3.080828327014939, −2.525733440387847, −1.600141099975423, −1.069011602916489, 0,
1.069011602916489, 1.600141099975423, 2.525733440387847, 3.080828327014939, 3.690261940469079, 4.377894676749654, 5.225887948371785, 5.524989889937297, 6.675732739253659, 6.885923774771646, 7.635246996145451, 8.034183777540630, 8.479051173307921, 8.989618197944799, 9.717431221622909, 10.02931778858304, 10.55442872545323, 11.24747605780093, 11.79173033279350, 12.09759596470051, 12.69063972957545, 13.40499189104834, 13.92687585849549, 14.38023180260625, 14.88765591028297