L(s) = 1 | − 2-s + 1.32·3-s + 4-s + 2.72·5-s − 1.32·6-s + 0.993·7-s − 8-s − 1.23·9-s − 2.72·10-s + 0.284·11-s + 1.32·12-s + 3.25·13-s − 0.993·14-s + 3.61·15-s + 16-s + 3.10·17-s + 1.23·18-s − 5.78·19-s + 2.72·20-s + 1.31·21-s − 0.284·22-s + 4.69·23-s − 1.32·24-s + 2.40·25-s − 3.25·26-s − 5.62·27-s + 0.993·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.766·3-s + 0.5·4-s + 1.21·5-s − 0.542·6-s + 0.375·7-s − 0.353·8-s − 0.412·9-s − 0.860·10-s + 0.0857·11-s + 0.383·12-s + 0.903·13-s − 0.265·14-s + 0.932·15-s + 0.250·16-s + 0.752·17-s + 0.291·18-s − 1.32·19-s + 0.608·20-s + 0.287·21-s − 0.0606·22-s + 0.979·23-s − 0.271·24-s + 0.480·25-s − 0.639·26-s − 1.08·27-s + 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.520077706\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.520077706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 0.993T + 7T^{2} \) |
| 11 | \( 1 - 0.284T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 5.78T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 9.57T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 8.95T + 43T^{2} \) |
| 47 | \( 1 + 0.959T + 47T^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 + 0.690T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 2.95T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−8.395860269118565234535197266041, −8.124701257013547168171331621317, −7.02987294412832847249835444223, −6.23420610404171374799456375782, −5.74015000271644855056104522131, −4.73099382114624789443787173610, −3.55012551521833730163127712261, −2.68374021993709548089334770822, −1.98277437540280329323208801062, −1.02441726800520707167819467626,
1.02441726800520707167819467626, 1.98277437540280329323208801062, 2.68374021993709548089334770822, 3.55012551521833730163127712261, 4.73099382114624789443787173610, 5.74015000271644855056104522131, 6.23420610404171374799456375782, 7.02987294412832847249835444223, 8.124701257013547168171331621317, 8.395860269118565234535197266041