L(s) = 1 | + 0.193·2-s − 2.48·3-s − 1.96·4-s − 0.481·6-s + 7-s − 0.768·8-s + 3.15·9-s + 1.19·11-s + 4.86·12-s + 2.96·13-s + 0.193·14-s + 3.77·16-s + 0.481·17-s + 0.612·18-s − 4.31·19-s − 2.48·21-s + 0.231·22-s − 23-s + 1.90·24-s + 0.574·26-s − 0.387·27-s − 1.96·28-s − 4.15·29-s + 10.2·31-s + 2.26·32-s − 2.96·33-s + 0.0933·34-s + ⋯ |
L(s) = 1 | + 0.137·2-s − 1.43·3-s − 0.981·4-s − 0.196·6-s + 0.377·7-s − 0.271·8-s + 1.05·9-s + 0.359·11-s + 1.40·12-s + 0.821·13-s + 0.0518·14-s + 0.943·16-s + 0.116·17-s + 0.144·18-s − 0.989·19-s − 0.541·21-s + 0.0493·22-s − 0.208·23-s + 0.389·24-s + 0.112·26-s − 0.0746·27-s − 0.370·28-s − 0.771·29-s + 1.83·31-s + 0.401·32-s − 0.515·33-s + 0.0160·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7981877048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7981877048\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 0.481T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 5.73T + 43T^{2} \) |
| 47 | \( 1 + 0.168T + 47T^{2} \) |
| 53 | \( 1 + 8.70T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 0.775T + 71T^{2} \) |
| 73 | \( 1 + 8.88T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 0.0933T + 97T^{2} \) |
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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
−8.454311412511253611039110500098, −7.79807169845103615919730102261, −6.63264666279038371027171497431, −6.11418879267833437266980956828, −5.51360607472759548503880074758, −4.61207763373888271781025574341, −4.28171840738164095283746032644, −3.18331514332770643042462514844, −1.58069807411594877123945982277, −0.57473501613142959468924361549,
0.57473501613142959468924361549, 1.58069807411594877123945982277, 3.18331514332770643042462514844, 4.28171840738164095283746032644, 4.61207763373888271781025574341, 5.51360607472759548503880074758, 6.11418879267833437266980956828, 6.63264666279038371027171497431, 7.79807169845103615919730102261, 8.454311412511253611039110500098