| L(s) = 1 | − 2.49·3-s + 3.75·5-s − 7-s + 3.23·9-s − 2.17·11-s − 3.12·13-s − 9.37·15-s + 17-s + 5.42·19-s + 2.49·21-s − 2.65·23-s + 9.09·25-s − 0.599·27-s − 6.42·29-s − 1.93·31-s + 5.42·33-s − 3.75·35-s + 9.89·37-s + 7.80·39-s − 5.75·41-s − 4.87·43-s + 12.1·45-s − 12.3·47-s + 49-s − 2.49·51-s + 0.635·53-s − 8.15·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 1.67·5-s − 0.377·7-s + 1.07·9-s − 0.654·11-s − 0.867·13-s − 2.42·15-s + 0.242·17-s + 1.24·19-s + 0.545·21-s − 0.554·23-s + 1.81·25-s − 0.115·27-s − 1.19·29-s − 0.348·31-s + 0.944·33-s − 0.634·35-s + 1.62·37-s + 1.25·39-s − 0.899·41-s − 0.743·43-s + 1.81·45-s − 1.80·47-s + 0.142·49-s − 0.349·51-s + 0.0873·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 - 3.75T + 5T^{2} \) |
| 11 | \( 1 + 2.17T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 0.635T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 8.06T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 6.15T + 73T^{2} \) |
| 79 | \( 1 - 0.358T + 79T^{2} \) |
| 83 | \( 1 - 1.64T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 + 4.12T + 97T^{2} \) |
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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
−7.948069107495714565975878127071, −7.18880774785927592035858599195, −6.29683885043946201489853221707, −5.96289257001448947119372395560, −5.12326520900979205586461116389, −4.92195656529938840023883681607, −3.33366814845751551746754572792, −2.32203404163882378370584488188, −1.34781187418814293042775889559, 0,
1.34781187418814293042775889559, 2.32203404163882378370584488188, 3.33366814845751551746754572792, 4.92195656529938840023883681607, 5.12326520900979205586461116389, 5.96289257001448947119372395560, 6.29683885043946201489853221707, 7.18880774785927592035858599195, 7.948069107495714565975878127071
