| L(s) = 1 | − 6.53·3-s − 20.9·5-s − 8.55·7-s + 15.6·9-s + 10.8·11-s − 54.7·13-s + 136.·15-s − 106.·17-s − 113.·19-s + 55.8·21-s + 112.·23-s + 312.·25-s + 74.1·27-s − 29·29-s + 102.·31-s − 70.5·33-s + 179.·35-s + 105.·37-s + 357.·39-s + 216.·41-s − 102.·43-s − 327.·45-s − 455.·47-s − 269.·49-s + 693.·51-s + 593.·53-s − 226.·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s − 1.87·5-s − 0.462·7-s + 0.579·9-s + 0.296·11-s − 1.16·13-s + 2.35·15-s − 1.51·17-s − 1.37·19-s + 0.580·21-s + 1.02·23-s + 2.50·25-s + 0.528·27-s − 0.185·29-s + 0.595·31-s − 0.372·33-s + 0.864·35-s + 0.469·37-s + 1.46·39-s + 0.826·41-s − 0.362·43-s − 1.08·45-s − 1.41·47-s − 0.786·49-s + 1.90·51-s + 1.53·53-s − 0.554·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 6.53T + 27T^{2} \) |
| 5 | \( 1 + 20.9T + 125T^{2} \) |
| 7 | \( 1 + 8.55T + 343T^{2} \) |
| 11 | \( 1 - 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 593.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 2.38T + 3.57e5T^{2} \) |
| 73 | \( 1 - 119.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 964.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 772.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−8.440765690909498154459944774714, −7.56167202618851635121212859500, −6.73266009911993471360029165062, −6.39325953197217509688509331664, −4.90366777865474039766962631973, −4.59799717042388793041952321457, −3.65371828702217050323256726083, −2.49214253770699080749450780737, −0.64796553037189631980619453841, 0,
0.64796553037189631980619453841, 2.49214253770699080749450780737, 3.65371828702217050323256726083, 4.59799717042388793041952321457, 4.90366777865474039766962631973, 6.39325953197217509688509331664, 6.73266009911993471360029165062, 7.56167202618851635121212859500, 8.440765690909498154459944774714
