| L(s) = 1 | + 3-s − 5-s − 3.37·7-s + 9-s + 5.37·11-s − 4.74·13-s − 15-s + 17-s − 3.37·19-s − 3.37·21-s + 4·23-s + 25-s + 27-s − 2.62·29-s − 0.744·31-s + 5.37·33-s + 3.37·35-s + 1.37·37-s − 4.74·39-s − 2.62·41-s + 8.74·43-s − 45-s + 5.37·47-s + 4.37·49-s + 51-s + 5.37·53-s − 5.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.27·7-s + 0.333·9-s + 1.61·11-s − 1.31·13-s − 0.258·15-s + 0.242·17-s − 0.773·19-s − 0.735·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.487·29-s − 0.133·31-s + 0.935·33-s + 0.570·35-s + 0.225·37-s − 0.759·39-s − 0.410·41-s + 1.33·43-s − 0.149·45-s + 0.783·47-s + 0.624·49-s + 0.140·51-s + 0.737·53-s − 0.724·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754941603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754941603\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 0.744T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.567597782778986287487319925928, −7.59003537019331205463974052315, −6.94271532286028424144070321998, −6.50599032397806591395725682787, −5.49628673903428657707892695728, −4.37916662337147099731910265346, −3.80498008728810601178675228378, −3.03279550900142602566353242778, −2.13302327016523550130671628821, −0.72224126411057492790127439304,
0.72224126411057492790127439304, 2.13302327016523550130671628821, 3.03279550900142602566353242778, 3.80498008728810601178675228378, 4.37916662337147099731910265346, 5.49628673903428657707892695728, 6.50599032397806591395725682787, 6.94271532286028424144070321998, 7.59003537019331205463974052315, 8.567597782778986287487319925928
