| L(s) = 1 | + 2.44·3-s + 1.41·7-s + 2.99·9-s − 2·11-s − 5.65·13-s − 4.89·17-s − 6·19-s + 3.46·21-s + 7.07·23-s − 6.92·29-s + 6.92·31-s − 4.89·33-s + 2.82·37-s − 13.8·39-s + 4·41-s − 2.44·43-s − 4.24·47-s − 5·49-s − 11.9·51-s − 14.6·57-s − 2·59-s − 3.46·61-s + 4.24·63-s + 2.44·67-s + 17.3·69-s + 6.92·71-s − 4.89·73-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 0.534·7-s + 0.999·9-s − 0.603·11-s − 1.56·13-s − 1.18·17-s − 1.37·19-s + 0.755·21-s + 1.47·23-s − 1.28·29-s + 1.24·31-s − 0.852·33-s + 0.464·37-s − 2.21·39-s + 0.624·41-s − 0.373·43-s − 0.618·47-s − 0.714·49-s − 1.68·51-s − 1.94·57-s − 0.260·59-s − 0.443·61-s + 0.534·63-s + 0.299·67-s + 2.08·69-s + 0.822·71-s − 0.573·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.891608599989156157662977403967, −7.13244102967098878929562522480, −6.52449418370677617909351706366, −5.31483315895409588978374581713, −4.64580091158980977231455924302, −4.05664473636801749738831776862, −2.85166243272533646525486940196, −2.50610155220190590941183348908, −1.70331265173492905390901899734, 0,
1.70331265173492905390901899734, 2.50610155220190590941183348908, 2.85166243272533646525486940196, 4.05664473636801749738831776862, 4.64580091158980977231455924302, 5.31483315895409588978374581713, 6.52449418370677617909351706366, 7.13244102967098878929562522480, 7.891608599989156157662977403967
