L(s) = 1 | + 0.704·2-s − 2.54·3-s − 1.50·4-s − 1.79·6-s + 7-s − 2.46·8-s + 3.48·9-s − 3.60·11-s + 3.82·12-s + 6.67·13-s + 0.704·14-s + 1.26·16-s − 7.65·17-s + 2.45·18-s + 6.35·19-s − 2.54·21-s − 2.54·22-s − 23-s + 6.28·24-s + 4.70·26-s − 1.23·27-s − 1.50·28-s − 5.44·29-s − 9.59·31-s + 5.83·32-s + 9.18·33-s − 5.38·34-s + ⋯ |
L(s) = 1 | + 0.498·2-s − 1.47·3-s − 0.751·4-s − 0.732·6-s + 0.377·7-s − 0.872·8-s + 1.16·9-s − 1.08·11-s + 1.10·12-s + 1.85·13-s + 0.188·14-s + 0.317·16-s − 1.85·17-s + 0.578·18-s + 1.45·19-s − 0.555·21-s − 0.541·22-s − 0.208·23-s + 1.28·24-s + 0.922·26-s − 0.237·27-s − 0.284·28-s − 1.01·29-s − 1.72·31-s + 1.03·32-s + 1.59·33-s − 0.924·34-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
\(L(1)\) |
\(\approx\) |
\(0.7097886626\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7097886626\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.704T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 - 6.67T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 29 | \( 1 + 5.44T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 - 6.80T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 - 0.0151T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 9.83T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 0.252T + 97T^{2} \) |
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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
−8.656479640496412632866230919611, −7.55117277145677751124838361071, −6.77954643716028666758694878235, −5.76974488684015251545860987032, −5.59237571568282184536600042828, −4.85857774267035546851124341939, −4.07327062143316614934113295221, −3.28531984733015290990440812607, −1.75998325462690671973805319830, −0.48634411864990291916183042018,
0.48634411864990291916183042018, 1.75998325462690671973805319830, 3.28531984733015290990440812607, 4.07327062143316614934113295221, 4.85857774267035546851124341939, 5.59237571568282184536600042828, 5.76974488684015251545860987032, 6.77954643716028666758694878235, 7.55117277145677751124838361071, 8.656479640496412632866230919611