| L(s) = 1 | − 1.39·3-s − 4.60·7-s − 1.04·9-s − 1.56·11-s + 2.60·13-s + 0.559·17-s + 1.16·19-s + 6.44·21-s + 23-s + 5.65·27-s − 3.17·29-s − 10.0·31-s + 2.19·33-s + 5.07·37-s − 3.63·39-s − 11.8·41-s + 2.76·43-s − 9.32·47-s + 14.2·49-s − 0.782·51-s − 5.54·53-s − 1.63·57-s − 3.84·59-s − 4.29·61-s + 4.79·63-s − 2.60·67-s − 1.39·69-s + ⋯ |
| L(s) = 1 | − 0.807·3-s − 1.74·7-s − 0.347·9-s − 0.472·11-s + 0.721·13-s + 0.135·17-s + 0.267·19-s + 1.40·21-s + 0.208·23-s + 1.08·27-s − 0.589·29-s − 1.80·31-s + 0.381·33-s + 0.834·37-s − 0.582·39-s − 1.84·41-s + 0.421·43-s − 1.35·47-s + 2.03·49-s − 0.109·51-s − 0.762·53-s − 0.216·57-s − 0.499·59-s − 0.550·61-s + 0.604·63-s − 0.318·67-s − 0.168·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3459721700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3459721700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 - 0.559T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 + 3.84T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 9.90T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 7.71T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−7.56620007713607048827387216052, −6.83117507212598488773396003708, −6.32289949651522798790573895906, −5.69987635844872964541888714591, −5.25076899345512340532273396993, −4.16904375768386603562002891628, −3.29259359538291606226640993214, −2.92597763013752115273812059982, −1.59498974261652318392737095422, −0.29003681398714223398442989766,
0.29003681398714223398442989766, 1.59498974261652318392737095422, 2.92597763013752115273812059982, 3.29259359538291606226640993214, 4.16904375768386603562002891628, 5.25076899345512340532273396993, 5.69987635844872964541888714591, 6.32289949651522798790573895906, 6.83117507212598488773396003708, 7.56620007713607048827387216052
