| L(s) = 1 | + 3·3-s − 27.0·7-s + 9·9-s + 1.66·11-s − 18.2·13-s − 39.2·17-s − 157.·19-s − 81.0·21-s − 115.·23-s + 27·27-s − 136.·29-s + 27.8·31-s + 5.00·33-s + 358.·37-s − 54.6·39-s − 30.3·41-s − 212.·43-s + 630.·47-s + 386.·49-s − 117.·51-s + 568.·53-s − 471.·57-s − 209.·59-s + 429.·61-s − 243.·63-s + 775.·67-s − 347.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.45·7-s + 0.333·9-s + 0.0457·11-s − 0.388·13-s − 0.559·17-s − 1.89·19-s − 0.841·21-s − 1.05·23-s + 0.192·27-s − 0.871·29-s + 0.161·31-s + 0.0263·33-s + 1.59·37-s − 0.224·39-s − 0.115·41-s − 0.754·43-s + 1.95·47-s + 1.12·49-s − 0.323·51-s + 1.47·53-s − 1.09·57-s − 0.461·59-s + 0.902·61-s − 0.486·63-s + 1.41·67-s − 0.606·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.288361134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288361134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 27.0T + 343T^{2} \) |
| 11 | \( 1 - 1.66T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 157.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 212.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 630.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 568.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 209.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 429.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 775.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 79.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 271.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 206.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 354.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 529.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
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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
−8.716881663477053260771972138617, −7.908239529727062824348233196891, −6.98334170352559591087873684436, −6.40324656121686158870528751464, −5.66194045677463047496604898315, −4.28078470127697118738510445242, −3.84423797489273073844873197571, −2.69348607220856346684307063583, −2.09219619472437112217962313234, −0.45741762921619387793858597710,
0.45741762921619387793858597710, 2.09219619472437112217962313234, 2.69348607220856346684307063583, 3.84423797489273073844873197571, 4.28078470127697118738510445242, 5.66194045677463047496604898315, 6.40324656121686158870528751464, 6.98334170352559591087873684436, 7.908239529727062824348233196891, 8.716881663477053260771972138617
