L(s) = 1 | − 0.481·3-s − 1.68·5-s − 3.55·7-s − 2.76·9-s − 0.0360·11-s + 1.38·13-s + 0.811·15-s − 17-s + 4.42·19-s + 1.71·21-s − 7.08·23-s − 2.16·25-s + 2.77·27-s − 4.48·29-s − 3.64·31-s + 0.0173·33-s + 5.99·35-s − 4.27·37-s − 0.668·39-s − 5.29·41-s + 1.97·43-s + 4.66·45-s − 12.5·47-s + 5.64·49-s + 0.481·51-s − 2.47·53-s + 0.0607·55-s + ⋯ |
L(s) = 1 | − 0.277·3-s − 0.753·5-s − 1.34·7-s − 0.922·9-s − 0.0108·11-s + 0.385·13-s + 0.209·15-s − 0.242·17-s + 1.01·19-s + 0.373·21-s − 1.47·23-s − 0.432·25-s + 0.534·27-s − 0.833·29-s − 0.654·31-s + 0.00302·33-s + 1.01·35-s − 0.702·37-s − 0.107·39-s − 0.826·41-s + 0.301·43-s + 0.695·45-s − 1.83·47-s + 0.806·49-s + 0.0674·51-s − 0.339·53-s + 0.00818·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2321656278\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2321656278\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 0.481T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 + 0.0360T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−7.891440497152971442836660575129, −7.07704565590504849238734004264, −6.41646515484485121049474511996, −5.80927255934164228595820169604, −5.19858583655196499758157137289, −4.07228679617339692707516326768, −3.48834811925240313726821756939, −2.94380250652792239887346287530, −1.74962782934109117697472166106, −0.23059759551856820675877948239,
0.23059759551856820675877948239, 1.74962782934109117697472166106, 2.94380250652792239887346287530, 3.48834811925240313726821756939, 4.07228679617339692707516326768, 5.19858583655196499758157137289, 5.80927255934164228595820169604, 6.41646515484485121049474511996, 7.07704565590504849238734004264, 7.891440497152971442836660575129