| L(s) = 1 | + 3-s − 5-s − 4.37·7-s + 9-s + 4.37·11-s − 13-s − 15-s + 0.372·17-s − 2·19-s − 4.37·21-s − 2.37·23-s + 25-s + 27-s − 2.74·29-s + 6.74·31-s + 4.37·33-s + 4.37·35-s + 4.37·37-s − 39-s − 0.372·41-s + 4.74·43-s − 45-s + 2.74·47-s + 12.1·49-s + 0.372·51-s − 7.62·53-s − 4.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.65·7-s + 0.333·9-s + 1.31·11-s − 0.277·13-s − 0.258·15-s + 0.0902·17-s − 0.458·19-s − 0.954·21-s − 0.494·23-s + 0.200·25-s + 0.192·27-s − 0.509·29-s + 1.21·31-s + 0.761·33-s + 0.739·35-s + 0.718·37-s − 0.160·39-s − 0.0581·41-s + 0.723·43-s − 0.149·45-s + 0.400·47-s + 1.73·49-s + 0.0521·51-s − 1.04·53-s − 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 0.372T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 1.25T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−7.65850080870521988519817595890, −6.94309427515614049687156616942, −6.38843838440150210586119062886, −5.81532768465000277304388660323, −4.44439984543876444517338429023, −3.98674035312219027409459734045, −3.20792184020664322468012143805, −2.56182931708867292820606685710, −1.28646576775690583960011701612, 0,
1.28646576775690583960011701612, 2.56182931708867292820606685710, 3.20792184020664322468012143805, 3.98674035312219027409459734045, 4.44439984543876444517338429023, 5.81532768465000277304388660323, 6.38843838440150210586119062886, 6.94309427515614049687156616942, 7.65850080870521988519817595890
