L(s) = 1 | + 2-s + 0.800·3-s + 4-s − 2.16·5-s + 0.800·6-s + 7-s + 8-s − 2.35·9-s − 2.16·10-s + 1.27·11-s + 0.800·12-s − 0.101·13-s + 14-s − 1.73·15-s + 16-s − 1.15·17-s − 2.35·18-s − 1.44·19-s − 2.16·20-s + 0.800·21-s + 1.27·22-s + 6.65·23-s + 0.800·24-s − 0.317·25-s − 0.101·26-s − 4.29·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.462·3-s + 0.5·4-s − 0.967·5-s + 0.326·6-s + 0.377·7-s + 0.353·8-s − 0.786·9-s − 0.684·10-s + 0.383·11-s + 0.231·12-s − 0.0281·13-s + 0.267·14-s − 0.447·15-s + 0.250·16-s − 0.279·17-s − 0.556·18-s − 0.331·19-s − 0.483·20-s + 0.174·21-s + 0.271·22-s + 1.38·23-s + 0.163·24-s − 0.0635·25-s − 0.0199·26-s − 0.825·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 0.800T + 3T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 + 0.101T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 + 0.966T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 + 9.55T + 53T^{2} \) |
| 59 | \( 1 - 0.694T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 2.60T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 - 8.78T + 73T^{2} \) |
| 79 | \( 1 - 7.84T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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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
−7.71739218640393330248163840237, −7.03597067833195919669903447601, −6.33861388197832246259940719418, −5.37306464267257064987505930304, −4.82287873340445542689165391588, −3.86616839695815395690922841118, −3.42701651596467886578816251612, −2.55423117326838654642404441221, −1.54766935049598944729150121907, 0,
1.54766935049598944729150121907, 2.55423117326838654642404441221, 3.42701651596467886578816251612, 3.86616839695815395690922841118, 4.82287873340445542689165391588, 5.37306464267257064987505930304, 6.33861388197832246259940719418, 7.03597067833195919669903447601, 7.71739218640393330248163840237