L(s) = 1 | + 2-s + 4-s + 2.64·5-s + 8-s + 2.64·10-s − 11-s + 4·13-s + 16-s − 3·17-s − 5.29·19-s + 2.64·20-s − 22-s + 2.64·23-s + 2.00·25-s + 4·26-s − 2·29-s + 4·31-s + 32-s − 3·34-s + 1.29·37-s − 5.29·38-s + 2.64·40-s + 9·41-s + 9.29·43-s − 44-s + 2.64·46-s − 3.93·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.18·5-s + 0.353·8-s + 0.836·10-s − 0.301·11-s + 1.10·13-s + 0.250·16-s − 0.727·17-s − 1.21·19-s + 0.591·20-s − 0.213·22-s + 0.551·23-s + 0.400·25-s + 0.784·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.212·37-s − 0.858·38-s + 0.418·40-s + 1.40·41-s + 1.41·43-s − 0.150·44-s + 0.390·46-s − 0.574·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.542064848\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.542064848\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.64132764128881286666852938513, −6.53219424750734388994783365266, −6.35613724994122282872227313561, −5.70504167314519613392573411436, −4.95189172853857840528190969235, −4.25570878043122212463295123386, −3.48759344223791640113922230250, −2.45854907330576115931757060400, −2.03349503542897661945235423405, −0.925283165064081036981958238165,
0.925283165064081036981958238165, 2.03349503542897661945235423405, 2.45854907330576115931757060400, 3.48759344223791640113922230250, 4.25570878043122212463295123386, 4.95189172853857840528190969235, 5.70504167314519613392573411436, 6.35613724994122282872227313561, 6.53219424750734388994783365266, 7.64132764128881286666852938513