| L(s) = 1 | + 2.73·3-s + 4.73i·7-s + 4.46·9-s − 3.46i·11-s + 3.46·13-s + 3.46i·17-s − 2i·19-s + 12.9i·21-s + 2.19i·23-s + 3.99·27-s + 2.53·31-s − 9.46i·33-s − 6·37-s + 9.46·39-s + 9.46·41-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 1.78i·7-s + 1.48·9-s − 1.04i·11-s + 0.960·13-s + 0.840i·17-s − 0.458i·19-s + 2.82i·21-s + 0.457i·23-s + 0.769·27-s + 0.455·31-s − 1.64i·33-s − 0.986·37-s + 1.51·39-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.100209061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.100209061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 - 2.19iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 0.928iT - 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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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
−9.072483355652518137207282732174, −8.655445026943286138868602417178, −8.379669971945716805501449986347, −7.34449687791853178782540678664, −6.09107836170262905874907901318, −5.62182683913409703939460422760, −4.19688731453951213127205559124, −3.22867147104805207472808013420, −2.64447588045411647119143316734, −1.60450781011561824536650431566,
1.10555075908559938036157228688, 2.25810805864677933439854880971, 3.42490921202614560283341593560, 4.01952647958183072133863350664, 4.78787332436533268920002738291, 6.38153684760235733368471546754, 7.35762463749472254843886373399, 7.55559431797087976561559266097, 8.535467329142109209293688671082, 9.233147016750791586293071994499
