| L(s) = 1 | − 3-s − 0.697·5-s − 7-s + 9-s + 5.60·11-s + 2.30·13-s + 0.697·15-s − 0.394·17-s − 0.394·19-s + 21-s − 23-s − 4.51·25-s − 27-s − 5.60·29-s − 3.60·31-s − 5.60·33-s + 0.697·35-s + 5.60·37-s − 2.30·39-s + 3.60·41-s + 4.30·43-s − 0.697·45-s − 4.60·47-s + 49-s + 0.394·51-s + 5.90·53-s − 3.90·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.311·5-s − 0.377·7-s + 0.333·9-s + 1.69·11-s + 0.638·13-s + 0.180·15-s − 0.0956·17-s − 0.0904·19-s + 0.218·21-s − 0.208·23-s − 0.902·25-s − 0.192·27-s − 1.04·29-s − 0.647·31-s − 0.975·33-s + 0.117·35-s + 0.921·37-s − 0.368·39-s + 0.563·41-s + 0.656·43-s − 0.103·45-s − 0.671·47-s + 0.142·49-s + 0.0552·51-s + 0.811·53-s − 0.526·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370205445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370205445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 0.697T + 5T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.394T + 17T^{2} \) |
| 19 | \( 1 + 0.394T + 19T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 5.90T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−9.394697835673315918453667476715, −8.444315702303884255589715784562, −7.54426355860624150909515381176, −6.65602138764097052109044486071, −6.15381859101883317712841434256, −5.26809472763970107795444237312, −3.93372846068522349014384366440, −3.77844288876216926084910495360, −2.07259857414161306009821603018, −0.830741143338204111284977411228,
0.830741143338204111284977411228, 2.07259857414161306009821603018, 3.77844288876216926084910495360, 3.93372846068522349014384366440, 5.26809472763970107795444237312, 6.15381859101883317712841434256, 6.65602138764097052109044486071, 7.54426355860624150909515381176, 8.444315702303884255589715784562, 9.394697835673315918453667476715
