L(s) = 1 | − 3-s + 5-s − 1.89·7-s + 9-s + 2.38·11-s − 4.38·13-s − 15-s + 0.761·17-s − 0.864·19-s + 1.89·21-s + 23-s + 25-s − 27-s + 8.76·29-s − 9.35·31-s − 2.38·33-s − 1.89·35-s − 0.103·37-s + 4.38·39-s + 9.53·41-s − 3.25·43-s + 45-s − 3.13·47-s − 3.40·49-s − 0.761·51-s + 8.01·53-s + 2.38·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.716·7-s + 0.333·9-s + 0.719·11-s − 1.21·13-s − 0.258·15-s + 0.184·17-s − 0.198·19-s + 0.413·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.62·29-s − 1.68·31-s − 0.415·33-s − 0.320·35-s − 0.0169·37-s + 0.702·39-s + 1.48·41-s − 0.495·43-s + 0.149·45-s − 0.457·47-s − 0.485·49-s − 0.106·51-s + 1.10·53-s + 0.321·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 0.761T + 17T^{2} \) |
| 19 | \( 1 + 0.864T + 19T^{2} \) |
| 29 | \( 1 - 8.76T + 29T^{2} \) |
| 31 | \( 1 + 9.35T + 31T^{2} \) |
| 37 | \( 1 + 0.103T + 37T^{2} \) |
| 41 | \( 1 - 9.53T + 41T^{2} \) |
| 43 | \( 1 + 3.25T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 6.55T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 0.0645T + 97T^{2} \) |
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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
−7.59379925235903249898669233788, −6.96735843547707639723555199673, −6.37888235372392597281462585962, −5.70983806368819028613914951523, −4.93530801048408249226542803945, −4.21205268443420562031152096122, −3.22222279573673447142068945571, −2.35964621995168022875338465977, −1.25584017134138386024805223605, 0,
1.25584017134138386024805223605, 2.35964621995168022875338465977, 3.22222279573673447142068945571, 4.21205268443420562031152096122, 4.93530801048408249226542803945, 5.70983806368819028613914951523, 6.37888235372392597281462585962, 6.96735843547707639723555199673, 7.59379925235903249898669233788