L(s) = 1 | + 3-s − 4.42·5-s − 7-s + 9-s + 2.55·11-s − 5.64·13-s − 4.42·15-s − 1.22·17-s + 19-s − 21-s + 7.95·23-s + 14.5·25-s + 27-s − 1.33·29-s + 5.95·31-s + 2.55·33-s + 4.42·35-s − 1.20·37-s − 5.64·39-s − 2·41-s + 0.551·43-s − 4.42·45-s + 8.72·47-s + 49-s − 1.22·51-s − 1.33·53-s − 11.2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.97·5-s − 0.377·7-s + 0.333·9-s + 0.769·11-s − 1.56·13-s − 1.14·15-s − 0.296·17-s + 0.229·19-s − 0.218·21-s + 1.65·23-s + 2.91·25-s + 0.192·27-s − 0.247·29-s + 1.06·31-s + 0.444·33-s + 0.748·35-s − 0.198·37-s − 0.904·39-s − 0.312·41-s + 0.0841·43-s − 0.659·45-s + 1.27·47-s + 0.142·49-s − 0.170·51-s − 0.182·53-s − 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 4.42T + 5T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 1.20T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 0.551T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 + 3.33T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 + 0.0494T + 79T^{2} \) |
| 83 | \( 1 + 7.96T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 1.64T + 97T^{2} \) |
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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
−7.69477125387864164974586907809, −6.97923218685578241516556398004, −6.79019241067168835295192863071, −5.30501650351753481400603351100, −4.50213785614058648759051459749, −4.08087982906211669939639565398, −3.13800496448750186935344340141, −2.69763847587083420433229180319, −1.12223219384984647949556425657, 0,
1.12223219384984647949556425657, 2.69763847587083420433229180319, 3.13800496448750186935344340141, 4.08087982906211669939639565398, 4.50213785614058648759051459749, 5.30501650351753481400603351100, 6.79019241067168835295192863071, 6.97923218685578241516556398004, 7.69477125387864164974586907809