| L(s) = 1 | + 1.30·3-s + 0.939·5-s + 2.17·7-s − 1.29·9-s − 1.01·11-s − 4.17·13-s + 1.22·15-s + 0.470·17-s − 3.44·19-s + 2.84·21-s − 0.962·23-s − 4.11·25-s − 5.60·27-s − 5.56·29-s − 7.56·31-s − 1.32·33-s + 2.04·35-s − 0.826·37-s − 5.45·39-s − 2.06·41-s − 1.18·43-s − 1.21·45-s + 4.23·47-s − 2.25·49-s + 0.614·51-s − 5.84·53-s − 0.953·55-s + ⋯ |
| L(s) = 1 | + 0.754·3-s + 0.420·5-s + 0.823·7-s − 0.431·9-s − 0.306·11-s − 1.15·13-s + 0.316·15-s + 0.114·17-s − 0.790·19-s + 0.621·21-s − 0.200·23-s − 0.823·25-s − 1.07·27-s − 1.03·29-s − 1.35·31-s − 0.230·33-s + 0.346·35-s − 0.135·37-s − 0.873·39-s − 0.322·41-s − 0.181·43-s − 0.181·45-s + 0.617·47-s − 0.321·49-s + 0.0860·51-s − 0.803·53-s − 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 0.939T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 0.470T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 0.962T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 7.56T + 31T^{2} \) |
| 37 | \( 1 + 0.826T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 0.246T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 + 5.59T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.88292271424097478803849849247, −7.72597514087883351251508893598, −6.69678742172372708416765806893, −5.66601828825842179160015144382, −5.18677256748106802315356699442, −4.20996963418094949446662806625, −3.33351379891357131724359007817, −2.26516396511601450366628664596, −1.86833244362070368105900127299, 0,
1.86833244362070368105900127299, 2.26516396511601450366628664596, 3.33351379891357131724359007817, 4.20996963418094949446662806625, 5.18677256748106802315356699442, 5.66601828825842179160015144382, 6.69678742172372708416765806893, 7.72597514087883351251508893598, 7.88292271424097478803849849247
