L(s) = 1 | + 2.73·2-s + 2.52·3-s + 5.48·4-s − 1.62·5-s + 6.89·6-s + 9.52·8-s + 3.36·9-s − 4.45·10-s − 1.21·11-s + 13.8·12-s − 3.49·13-s − 4.10·15-s + 15.0·16-s + 3.59·17-s + 9.19·18-s − 0.898·19-s − 8.92·20-s − 3.32·22-s + 8.47·23-s + 24.0·24-s − 2.35·25-s − 9.55·26-s + 0.915·27-s − 7.30·29-s − 11.2·30-s − 10.0·31-s + 22.1·32-s + ⋯ |
L(s) = 1 | + 1.93·2-s + 1.45·3-s + 2.74·4-s − 0.727·5-s + 2.81·6-s + 3.36·8-s + 1.12·9-s − 1.40·10-s − 0.367·11-s + 3.99·12-s − 0.969·13-s − 1.06·15-s + 3.76·16-s + 0.871·17-s + 2.16·18-s − 0.206·19-s − 1.99·20-s − 0.709·22-s + 1.76·23-s + 4.90·24-s − 0.470·25-s − 1.87·26-s + 0.176·27-s − 1.35·29-s − 2.05·30-s − 1.80·31-s + 3.92·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.452160500\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.452160500\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 + 0.898T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.346T + 37T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 - 0.195T + 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 - 0.144T + 61T^{2} \) |
| 67 | \( 1 + 2.58T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.064666781377765048182280509134, −7.970046398786115320076478256757, −7.41876820772617765749258739815, −6.98726271147623394542212952382, −5.59891529715385714696554627049, −5.00595868413010612809896176702, −3.93885175419994342214601561100, −3.46784143959195869830516684933, −2.70950414519358876546896179641, −1.85179962762195688291025639052,
1.85179962762195688291025639052, 2.70950414519358876546896179641, 3.46784143959195869830516684933, 3.93885175419994342214601561100, 5.00595868413010612809896176702, 5.59891529715385714696554627049, 6.98726271147623394542212952382, 7.41876820772617765749258739815, 7.970046398786115320076478256757, 9.064666781377765048182280509134