L(s) = 1 | + 0.346·2-s + 3-s − 1.87·4-s − 5-s + 0.346·6-s + 7-s − 1.34·8-s + 9-s − 0.346·10-s + 11-s − 1.87·12-s + 3.69·13-s + 0.346·14-s − 15-s + 3.29·16-s − 8.05·17-s + 0.346·18-s + 4.87·19-s + 1.87·20-s + 21-s + 0.346·22-s + 3.57·23-s − 1.34·24-s + 25-s + 1.27·26-s + 27-s − 1.87·28-s + ⋯ |
L(s) = 1 | + 0.244·2-s + 0.577·3-s − 0.939·4-s − 0.447·5-s + 0.141·6-s + 0.377·7-s − 0.475·8-s + 0.333·9-s − 0.109·10-s + 0.301·11-s − 0.542·12-s + 1.02·13-s + 0.0925·14-s − 0.258·15-s + 0.823·16-s − 1.95·17-s + 0.0816·18-s + 1.11·19-s + 0.420·20-s + 0.218·21-s + 0.0738·22-s + 0.744·23-s − 0.274·24-s + 0.200·25-s + 0.250·26-s + 0.192·27-s − 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.795342286\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795342286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 0.346T + 2T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 + 8.05T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 0.383T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 + 2.29T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 + 8.25T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.459244819989189425436147133026, −8.926417134535030872577459539112, −8.376132601046391870242404745401, −7.44788414101662681584795753486, −6.47250794486271379068119098779, −5.35140043162231701887602198012, −4.37737206667780255461239884531, −3.82660749712149201864831250705, −2.66874038196371146122925188530, −1.01025351930272324010087641915,
1.01025351930272324010087641915, 2.66874038196371146122925188530, 3.82660749712149201864831250705, 4.37737206667780255461239884531, 5.35140043162231701887602198012, 6.47250794486271379068119098779, 7.44788414101662681584795753486, 8.376132601046391870242404745401, 8.926417134535030872577459539112, 9.459244819989189425436147133026