| L(s) = 1 | − 0.301·3-s + 3.72·5-s + 3.06·7-s − 2.90·9-s − 1.17·11-s − 4.64·13-s − 1.12·15-s − 1.45·17-s + 3.85·19-s − 0.924·21-s + 5.40·23-s + 8.88·25-s + 1.78·27-s + 5.63·29-s + 4.49·31-s + 0.353·33-s + 11.4·35-s − 7.33·37-s + 1.40·39-s − 11.3·41-s − 1.68·43-s − 10.8·45-s + 0.152·47-s + 2.41·49-s + 0.437·51-s + 6.85·53-s − 4.37·55-s + ⋯ |
| L(s) = 1 | − 0.174·3-s + 1.66·5-s + 1.15·7-s − 0.969·9-s − 0.353·11-s − 1.28·13-s − 0.289·15-s − 0.352·17-s + 0.884·19-s − 0.201·21-s + 1.12·23-s + 1.77·25-s + 0.342·27-s + 1.04·29-s + 0.807·31-s + 0.0615·33-s + 1.93·35-s − 1.20·37-s + 0.224·39-s − 1.77·41-s − 0.256·43-s − 1.61·45-s + 0.0222·47-s + 0.344·49-s + 0.0612·51-s + 0.942·53-s − 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.846265954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.846265954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.301T + 3T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 - 0.152T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 1.33T + 83T^{2} \) |
| 89 | \( 1 - 6.98T + 89T^{2} \) |
| 97 | \( 1 - 7.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.902761671245359251218747984725, −6.95928635072662840988923753240, −6.50133972167459150496839289070, −5.43373189647154706993725818041, −5.17249890128122742898028829635, −4.77863056085056870582307148750, −3.23513662080069816008480672883, −2.48609496519807030472948893831, −1.91685750094414128783594633594, −0.849910793927187319502962891867,
0.849910793927187319502962891867, 1.91685750094414128783594633594, 2.48609496519807030472948893831, 3.23513662080069816008480672883, 4.77863056085056870582307148750, 5.17249890128122742898028829635, 5.43373189647154706993725818041, 6.50133972167459150496839289070, 6.95928635072662840988923753240, 7.902761671245359251218747984725
