L(s) = 1 | + 2.09·3-s − 5-s + 1.50·7-s + 1.40·9-s + 2.47·11-s − 2.17·13-s − 2.09·15-s − 6.53·17-s + 5.50·19-s + 3.16·21-s + 3.12·23-s + 25-s − 3.34·27-s + 2.41·29-s − 4.51·31-s + 5.20·33-s − 1.50·35-s + 10.1·37-s − 4.57·39-s + 10.7·41-s + 0.855·43-s − 1.40·45-s + 0.0391·47-s − 4.72·49-s − 13.7·51-s + 10.7·53-s − 2.47·55-s + ⋯ |
L(s) = 1 | + 1.21·3-s − 0.447·5-s + 0.569·7-s + 0.469·9-s + 0.747·11-s − 0.604·13-s − 0.542·15-s − 1.58·17-s + 1.26·19-s + 0.690·21-s + 0.652·23-s + 0.200·25-s − 0.643·27-s + 0.448·29-s − 0.810·31-s + 0.905·33-s − 0.254·35-s + 1.66·37-s − 0.732·39-s + 1.67·41-s + 0.130·43-s − 0.209·45-s + 0.00570·47-s − 0.675·49-s − 1.92·51-s + 1.47·53-s − 0.334·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.111391154\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.111391154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 2.09T + 3T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.855T + 43T^{2} \) |
| 47 | \( 1 - 0.0391T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 - 0.279T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 3.84T + 83T^{2} \) |
| 89 | \( 1 - 4.53T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.033557064395611897758606161004, −7.52736011831552770238202204692, −6.93601525646164512354663706620, −6.00239182984036495090917900747, −4.97873828419013819521198483542, −4.31162164249214878202159964210, −3.61567325230445914715681106793, −2.71982458558563341758364341923, −2.09855917572735404326542290917, −0.884811446252608357616756605657,
0.884811446252608357616756605657, 2.09855917572735404326542290917, 2.71982458558563341758364341923, 3.61567325230445914715681106793, 4.31162164249214878202159964210, 4.97873828419013819521198483542, 6.00239182984036495090917900747, 6.93601525646164512354663706620, 7.52736011831552770238202204692, 8.033557064395611897758606161004