L(s) = 1 | + 0.0800·3-s + 0.442·5-s + 2.60·7-s − 2.99·9-s − 2.05·11-s + 2.33·13-s + 0.0354·15-s + 2.04·17-s − 19-s + 0.208·21-s − 6.75·23-s − 4.80·25-s − 0.479·27-s − 0.161·29-s − 7.92·31-s − 0.164·33-s + 1.15·35-s + 3.74·37-s + 0.187·39-s − 5.17·41-s − 4.07·43-s − 1.32·45-s + 2.84·47-s − 0.213·49-s + 0.163·51-s + 53-s − 0.909·55-s + ⋯ |
L(s) = 1 | + 0.0462·3-s + 0.198·5-s + 0.984·7-s − 0.997·9-s − 0.619·11-s + 0.648·13-s + 0.00915·15-s + 0.494·17-s − 0.229·19-s + 0.0455·21-s − 1.40·23-s − 0.960·25-s − 0.0923·27-s − 0.0299·29-s − 1.42·31-s − 0.0286·33-s + 0.194·35-s + 0.615·37-s + 0.0299·39-s − 0.807·41-s − 0.621·43-s − 0.197·45-s + 0.415·47-s − 0.0304·49-s + 0.0228·51-s + 0.137·53-s − 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 - 0.0800T + 3T^{2} \) |
| 5 | \( 1 - 0.442T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 - 2.33T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 + 0.161T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 - 3.74T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 4.07T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.25T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.031675795909423293537657615703, −7.65910858744341885570813218049, −6.48369071033616105758739100950, −5.67712230865080370706950935751, −5.31525065232321712725580273921, −4.23018258424691325986437697642, −3.43296991264731113104034989191, −2.35774962156931810740124080077, −1.57461544171146835514338887297, 0,
1.57461544171146835514338887297, 2.35774962156931810740124080077, 3.43296991264731113104034989191, 4.23018258424691325986437697642, 5.31525065232321712725580273921, 5.67712230865080370706950935751, 6.48369071033616105758739100950, 7.65910858744341885570813218049, 8.031675795909423293537657615703