| L(s) = 1 | − 1.96·3-s − 1.29·5-s − 5.21·7-s + 0.880·9-s − 5.81·11-s + 4.89·13-s + 2.55·15-s + 5.22·17-s − 19-s + 10.2·21-s + 4.72·23-s − 3.31·25-s + 4.17·27-s + 6.72·29-s + 3.44·31-s + 11.4·33-s + 6.76·35-s + 1.61·37-s − 9.63·39-s − 4.72·41-s − 1.03·43-s − 1.14·45-s + 1.95·47-s + 20.2·49-s − 10.2·51-s + 53-s + 7.54·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s − 0.579·5-s − 1.97·7-s + 0.293·9-s − 1.75·11-s + 1.35·13-s + 0.659·15-s + 1.26·17-s − 0.229·19-s + 2.24·21-s + 0.985·23-s − 0.663·25-s + 0.803·27-s + 1.24·29-s + 0.618·31-s + 1.99·33-s + 1.14·35-s + 0.264·37-s − 1.54·39-s − 0.738·41-s − 0.157·43-s − 0.170·45-s + 0.285·47-s + 2.88·49-s − 1.44·51-s + 0.137·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 + 5.21T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 1.95T + 47T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 + 0.689T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 - 2.39T + 79T^{2} \) |
| 83 | \( 1 - 0.0104T + 83T^{2} \) |
| 89 | \( 1 + 8.42T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.075621732708207718802355706830, −7.16107041574353148764352309720, −6.47426116967909488213872387844, −5.84622270951993273612501378210, −5.36297173782492128379624358637, −4.29801683017091447681615729615, −3.23177481895192636118933899549, −2.87898522712951977123692943532, −0.893562301369399154648904062134, 0,
0.893562301369399154648904062134, 2.87898522712951977123692943532, 3.23177481895192636118933899549, 4.29801683017091447681615729615, 5.36297173782492128379624358637, 5.84622270951993273612501378210, 6.47426116967909488213872387844, 7.16107041574353148764352309720, 8.075621732708207718802355706830
