| L(s) = 1 | − 3-s + 4.38·5-s + 3.18·7-s + 9-s − 2.48·11-s − 5.09·13-s − 4.38·15-s − 7.14·17-s − 3.01·19-s − 3.18·21-s + 5.68·23-s + 14.1·25-s − 27-s − 7.79·29-s − 9.08·31-s + 2.48·33-s + 13.9·35-s − 4.53·37-s + 5.09·39-s − 3.01·41-s + 5.11·43-s + 4.38·45-s − 7.32·47-s + 3.13·49-s + 7.14·51-s − 13.7·53-s − 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.95·5-s + 1.20·7-s + 0.333·9-s − 0.749·11-s − 1.41·13-s − 1.13·15-s − 1.73·17-s − 0.690·19-s − 0.694·21-s + 1.18·23-s + 2.83·25-s − 0.192·27-s − 1.44·29-s − 1.63·31-s + 0.432·33-s + 2.35·35-s − 0.745·37-s + 0.815·39-s − 0.470·41-s + 0.779·43-s + 0.652·45-s − 1.06·47-s + 0.447·49-s + 1.00·51-s − 1.89·53-s − 1.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 4.38T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 7.14T + 17T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 0.195T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 0.215T + 83T^{2} \) |
| 89 | \( 1 + 9.49T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
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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
−7.45372001680577669973736141340, −7.03315006429823420740438578647, −6.17353970446707244523329500958, −5.46982935340836091962868620999, −4.95529599582044107224934530488, −4.54482408971241547133772026712, −2.90352328611162303209407513515, −1.93326887926170549995975639334, −1.78287476174293645150667559259, 0,
1.78287476174293645150667559259, 1.93326887926170549995975639334, 2.90352328611162303209407513515, 4.54482408971241547133772026712, 4.95529599582044107224934530488, 5.46982935340836091962868620999, 6.17353970446707244523329500958, 7.03315006429823420740438578647, 7.45372001680577669973736141340
