| L(s) = 1 | − 0.448·2-s + 5.38·3-s − 7.79·4-s + 8.53·5-s − 2.41·6-s + 7.08·8-s + 2.04·9-s − 3.82·10-s + 57.0·11-s − 42.0·12-s − 0.733·13-s + 45.9·15-s + 59.2·16-s − 97.4·17-s − 0.916·18-s − 49.7·19-s − 66.5·20-s − 25.6·22-s − 31.1·23-s + 38.1·24-s − 52.2·25-s + 0.328·26-s − 134.·27-s − 226.·29-s − 20.6·30-s − 31·31-s − 83.2·32-s + ⋯ |
| L(s) = 1 | − 0.158·2-s + 1.03·3-s − 0.974·4-s + 0.763·5-s − 0.164·6-s + 0.313·8-s + 0.0756·9-s − 0.121·10-s + 1.56·11-s − 1.01·12-s − 0.0156·13-s + 0.791·15-s + 0.925·16-s − 1.39·17-s − 0.0120·18-s − 0.600·19-s − 0.743·20-s − 0.248·22-s − 0.282·23-s + 0.324·24-s − 0.417·25-s + 0.00248·26-s − 0.958·27-s − 1.44·29-s − 0.125·30-s − 0.179·31-s − 0.459·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 31 | \( 1 + 31T \) |
| good | 2 | \( 1 + 0.448T + 8T^{2} \) |
| 3 | \( 1 - 5.38T + 27T^{2} \) |
| 5 | \( 1 - 8.53T + 125T^{2} \) |
| 11 | \( 1 - 57.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.733T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 230.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.47T + 1.03e5T^{2} \) |
| 53 | \( 1 - 171.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 233.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 648.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 694.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 288.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 959.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 17.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 833.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 608.T + 9.12e5T^{2} \) |
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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
−8.944646831593161619849812844277, −8.239453052604116391336713933878, −7.19932372000665710870206125247, −6.24749935271218261061688073248, −5.40017439625162383397275939940, −4.13627428493370363041818336431, −3.75095391737908361128740601745, −2.34819377909372135409126452890, −1.53824740775159975643266190129, 0,
1.53824740775159975643266190129, 2.34819377909372135409126452890, 3.75095391737908361128740601745, 4.13627428493370363041818336431, 5.40017439625162383397275939940, 6.24749935271218261061688073248, 7.19932372000665710870206125247, 8.239453052604116391336713933878, 8.944646831593161619849812844277
