| L(s) = 1 | − 0.111·3-s + 11.3·5-s − 3.28·7-s − 26.9·9-s − 48.3·11-s + 21.9·13-s − 1.26·15-s + 19.9·17-s + 116.·19-s + 0.367·21-s + 191.·23-s + 3.41·25-s + 6.03·27-s + 32.4·29-s + 209.·31-s + 5.40·33-s − 37.2·35-s + 37·37-s − 2.45·39-s + 128.·41-s − 299.·43-s − 305.·45-s + 369.·47-s − 332.·49-s − 2.23·51-s + 230.·53-s − 547.·55-s + ⋯ |
| L(s) = 1 | − 0.0215·3-s + 1.01·5-s − 0.177·7-s − 0.999·9-s − 1.32·11-s + 0.467·13-s − 0.0218·15-s + 0.284·17-s + 1.40·19-s + 0.00382·21-s + 1.73·23-s + 0.0272·25-s + 0.0430·27-s + 0.207·29-s + 1.21·31-s + 0.0285·33-s − 0.179·35-s + 0.164·37-s − 0.0100·39-s + 0.488·41-s − 1.06·43-s − 1.01·45-s + 1.14·47-s − 0.968·49-s − 0.00612·51-s + 0.597·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.127614804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127614804\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 + 0.111T + 27T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 + 3.28T + 343T^{2} \) |
| 11 | \( 1 + 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 32.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 369.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 230.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 49.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 907.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 67.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 661.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 30.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 207.T + 9.12e5T^{2} \) |
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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
−10.22349383604481967023823757897, −9.484601290160699483362675861633, −8.582390987524394713611534518906, −7.68532125881826944132178357692, −6.53767586744953499800995690275, −5.55888222246461570870793251257, −5.07327392021996238836596682744, −3.23990746027846512734428895591, −2.47806035829924153553369890662, −0.881459108141275358359477695295,
0.881459108141275358359477695295, 2.47806035829924153553369890662, 3.23990746027846512734428895591, 5.07327392021996238836596682744, 5.55888222246461570870793251257, 6.53767586744953499800995690275, 7.68532125881826944132178357692, 8.582390987524394713611534518906, 9.484601290160699483362675861633, 10.22349383604481967023823757897
