L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 1.09·7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 3.09·13-s + 1.09·14-s − 15-s + 16-s − 17-s + 18-s + 3.09·19-s − 20-s + 1.09·21-s + 22-s + 0.900·23-s + 24-s + 25-s + 3.09·26-s + 27-s + 1.09·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.415·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.859·13-s + 0.293·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.711·19-s − 0.223·20-s + 0.239·21-s + 0.213·22-s + 0.187·23-s + 0.204·24-s + 0.200·25-s + 0.607·26-s + 0.192·27-s + 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.332634470\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.332634470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 1.09T + 7T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 - 0.900T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.79T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 9.07T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.923573360561934827678640771585, −7.56447448365561199755908665686, −6.64965320479290704749321415034, −6.02829963542980409077747937999, −5.06885716971051568962391608242, −4.41756635564785506432414871330, −3.64908752154521235142016319175, −3.05256528266573664699315855774, −2.00860823532887615138429327281, −1.03370832383641508692050898394,
1.03370832383641508692050898394, 2.00860823532887615138429327281, 3.05256528266573664699315855774, 3.64908752154521235142016319175, 4.41756635564785506432414871330, 5.06885716971051568962391608242, 6.02829963542980409077747937999, 6.64965320479290704749321415034, 7.56447448365561199755908665686, 7.923573360561934827678640771585