| L(s) = 1 | + 0.760·2-s − 1.12·3-s − 1.42·4-s − 0.859·6-s − 2.53·7-s − 2.60·8-s − 1.72·9-s − 2.06·11-s + 1.60·12-s + 4.79·13-s − 1.93·14-s + 0.862·16-s + 1.95·17-s − 1.31·18-s − 1.03·19-s + 2.86·21-s − 1.56·22-s − 2.68·23-s + 2.93·24-s + 3.64·26-s + 5.33·27-s + 3.60·28-s + 5.84·29-s + 7.76·31-s + 5.86·32-s + 2.32·33-s + 1.48·34-s + ⋯ |
| L(s) = 1 | + 0.537·2-s − 0.652·3-s − 0.710·4-s − 0.350·6-s − 0.959·7-s − 0.920·8-s − 0.574·9-s − 0.621·11-s + 0.463·12-s + 1.33·13-s − 0.516·14-s + 0.215·16-s + 0.473·17-s − 0.309·18-s − 0.238·19-s + 0.625·21-s − 0.334·22-s − 0.560·23-s + 0.600·24-s + 0.715·26-s + 1.02·27-s + 0.681·28-s + 1.08·29-s + 1.39·31-s + 1.03·32-s + 0.405·33-s + 0.254·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.760T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 - 1.95T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 4.22T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 + 6.51T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 + 0.236T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.79584266241562390745645959933, −6.62776323436728743740869906587, −6.10627805106515382572373190128, −5.67839433209263180898756235522, −4.89011032304941008466087490594, −4.10084631581931848432804656463, −3.29651959529989183458958363096, −2.70272826225156913945800475184, −1.00100909570957711724297273043, 0,
1.00100909570957711724297273043, 2.70272826225156913945800475184, 3.29651959529989183458958363096, 4.10084631581931848432804656463, 4.89011032304941008466087490594, 5.67839433209263180898756235522, 6.10627805106515382572373190128, 6.62776323436728743740869906587, 7.79584266241562390745645959933
