| L(s) = 1 | − 1.46·3-s + 4.92·7-s − 0.866·9-s + 2.92·11-s − 3.86·13-s − 3.18·17-s − 8.10·19-s − 7.18·21-s − 23-s + 5.64·27-s − 9.70·29-s − 1.48·31-s − 4.26·33-s − 4.92·37-s + 5.64·39-s + 2.91·41-s + 3.73·43-s + 11.1·47-s + 17.2·49-s + 4.65·51-s + 5.29·53-s + 11.8·57-s − 9.65·59-s − 9.73·61-s − 4.26·63-s − 15.4·67-s + 1.46·69-s + ⋯ |
| L(s) = 1 | − 0.843·3-s + 1.85·7-s − 0.288·9-s + 0.880·11-s − 1.07·13-s − 0.772·17-s − 1.86·19-s − 1.56·21-s − 0.208·23-s + 1.08·27-s − 1.80·29-s − 0.266·31-s − 0.742·33-s − 0.809·37-s + 0.904·39-s + 0.455·41-s + 0.569·43-s + 1.62·47-s + 2.45·49-s + 0.651·51-s + 0.727·53-s + 1.56·57-s − 1.25·59-s − 1.24·61-s − 0.537·63-s − 1.88·67-s + 0.175·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 8.10T + 19T^{2} \) |
| 29 | \( 1 + 9.70T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 0.812T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 - 0.266T + 97T^{2} \) |
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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
−8.784773708594681976725670660654, −7.75951433706526903475884415102, −7.13006761147834870615376146754, −6.12000744902824062889527034354, −5.47359337619895615797619993667, −4.57472758781198945227058904944, −4.14884968201246521372164031333, −2.39557735051727559153836807214, −1.59928061157995782030515106388, 0,
1.59928061157995782030515106388, 2.39557735051727559153836807214, 4.14884968201246521372164031333, 4.57472758781198945227058904944, 5.47359337619895615797619993667, 6.12000744902824062889527034354, 7.13006761147834870615376146754, 7.75951433706526903475884415102, 8.784773708594681976725670660654
