L(s) = 1 | − 0.193·2-s − 3-s − 1.96·4-s + 0.930·5-s + 0.193·6-s + 3.24·7-s + 0.765·8-s + 9-s − 0.179·10-s − 0.492·11-s + 1.96·12-s + 0.350·13-s − 0.625·14-s − 0.930·15-s + 3.77·16-s − 17-s − 0.193·18-s + 0.0213·19-s − 1.82·20-s − 3.24·21-s + 0.0951·22-s − 7.67·23-s − 0.765·24-s − 4.13·25-s − 0.0676·26-s − 27-s − 6.35·28-s + ⋯ |
L(s) = 1 | − 0.136·2-s − 0.577·3-s − 0.981·4-s + 0.416·5-s + 0.0788·6-s + 1.22·7-s + 0.270·8-s + 0.333·9-s − 0.0568·10-s − 0.148·11-s + 0.566·12-s + 0.0971·13-s − 0.167·14-s − 0.240·15-s + 0.944·16-s − 0.242·17-s − 0.0455·18-s + 0.00490·19-s − 0.408·20-s − 0.707·21-s + 0.0202·22-s − 1.60·23-s − 0.156·24-s − 0.826·25-s − 0.0132·26-s − 0.192·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 5 | \( 1 - 0.930T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 0.492T + 11T^{2} \) |
| 13 | \( 1 - 0.350T + 13T^{2} \) |
| 19 | \( 1 - 0.0213T + 19T^{2} \) |
| 23 | \( 1 + 7.67T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 - 0.961T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 + 6.17T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.91T + 59T^{2} \) |
| 61 | \( 1 - 0.289T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + 0.851T + 73T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 2.22T + 97T^{2} \) |
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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
−8.258023970907782660684165852310, −7.49820207861235429284374638753, −6.52192423404416746550728640940, −5.57496447338063675315765710193, −5.20813329600295929002010058831, −4.35106724532815930736876380265, −3.73334556514516051261091613233, −2.17360102342191557734944470630, −1.34224369825606489527226759971, 0,
1.34224369825606489527226759971, 2.17360102342191557734944470630, 3.73334556514516051261091613233, 4.35106724532815930736876380265, 5.20813329600295929002010058831, 5.57496447338063675315765710193, 6.52192423404416746550728640940, 7.49820207861235429284374638753, 8.258023970907782660684165852310