L(s) = 1 | + 3-s + 5-s − 1.40·7-s + 9-s + 3.19·11-s − 7.12·13-s + 15-s + 5.07·17-s − 5.01·19-s − 1.40·21-s − 3.97·23-s + 25-s + 27-s − 9.15·29-s − 7.41·31-s + 3.19·33-s − 1.40·35-s − 5.97·37-s − 7.12·39-s + 0.574·41-s − 0.608·43-s + 45-s + 2.70·47-s − 5.01·49-s + 5.07·51-s + 3.64·53-s + 3.19·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.532·7-s + 0.333·9-s + 0.962·11-s − 1.97·13-s + 0.258·15-s + 1.23·17-s − 1.15·19-s − 0.307·21-s − 0.829·23-s + 0.200·25-s + 0.192·27-s − 1.70·29-s − 1.33·31-s + 0.555·33-s − 0.238·35-s − 0.982·37-s − 1.14·39-s + 0.0896·41-s − 0.0928·43-s + 0.149·45-s + 0.394·47-s − 0.716·49-s + 0.711·51-s + 0.500·53-s + 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 9.15T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 0.574T + 41T^{2} \) |
| 43 | \( 1 + 0.608T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 + 0.942T + 61T^{2} \) |
| 71 | \( 1 - 4.36T + 71T^{2} \) |
| 73 | \( 1 + 9.56T + 73T^{2} \) |
| 79 | \( 1 + 6.26T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 7.08T + 97T^{2} \) |
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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
−8.008004080651893172003956518427, −7.33397127875288911481067103542, −6.76478545684822891746811015603, −5.83158315022303789648151257065, −5.14188644371584092288033968879, −4.08256926417101592944951646741, −3.44553612205098726120072835516, −2.38016350821095152849849696507, −1.69889195859689866806536480943, 0,
1.69889195859689866806536480943, 2.38016350821095152849849696507, 3.44553612205098726120072835516, 4.08256926417101592944951646741, 5.14188644371584092288033968879, 5.83158315022303789648151257065, 6.76478545684822891746811015603, 7.33397127875288911481067103542, 8.008004080651893172003956518427