L(s) = 1 | − 0.554·3-s + 3.11·5-s + 7-s − 2.69·9-s − 11-s + 13-s − 1.72·15-s − 0.195·17-s + 2.90·19-s − 0.554·21-s + 2.28·23-s + 4.69·25-s + 3.15·27-s − 2.31·29-s − 4.38·31-s + 0.554·33-s + 3.11·35-s + 1.83·37-s − 0.554·39-s + 12.6·41-s + 0.100·43-s − 8.38·45-s − 11.4·47-s + 49-s + 0.108·51-s + 9.79·53-s − 3.11·55-s + ⋯ |
L(s) = 1 | − 0.320·3-s + 1.39·5-s + 0.377·7-s − 0.897·9-s − 0.301·11-s + 0.277·13-s − 0.446·15-s − 0.0473·17-s + 0.667·19-s − 0.121·21-s + 0.476·23-s + 0.939·25-s + 0.607·27-s − 0.430·29-s − 0.787·31-s + 0.0965·33-s + 0.526·35-s + 0.301·37-s − 0.0888·39-s + 1.97·41-s + 0.0153·43-s − 1.24·45-s − 1.66·47-s + 0.142·49-s + 0.0151·51-s + 1.34·53-s − 0.419·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.216796402\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.216796402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 0.554T + 3T^{2} \) |
| 5 | \( 1 - 3.11T + 5T^{2} \) |
| 17 | \( 1 + 0.195T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 - 0.100T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 - 9.20T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.0580T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.518836390401322016888222058809, −7.75924976060804155286431185415, −6.82539904265649564396244643709, −6.09087250773445040757188907251, −5.43976309047423920549547466982, −5.08880847334470078826666687397, −3.81397800930180219978553015021, −2.74576932846564588632036607666, −2.03536330982297771478332068973, −0.881845823439036896017926428357,
0.881845823439036896017926428357, 2.03536330982297771478332068973, 2.74576932846564588632036607666, 3.81397800930180219978553015021, 5.08880847334470078826666687397, 5.43976309047423920549547466982, 6.09087250773445040757188907251, 6.82539904265649564396244643709, 7.75924976060804155286431185415, 8.518836390401322016888222058809