| L(s) = 1 | + 1.62·2-s − 0.148·3-s + 0.650·4-s + 1.34·5-s − 0.242·6-s − 2.21·7-s − 2.19·8-s − 2.97·9-s + 2.19·10-s + 5.82·11-s − 0.0968·12-s − 0.647·13-s − 3.60·14-s − 0.201·15-s − 4.87·16-s − 5.33·17-s − 4.84·18-s + 2.89·19-s + 0.877·20-s + 0.330·21-s + 9.48·22-s + 2.66·23-s + 0.327·24-s − 3.17·25-s − 1.05·26-s + 0.890·27-s − 1.44·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.0859·3-s + 0.325·4-s + 0.603·5-s − 0.0989·6-s − 0.837·7-s − 0.776·8-s − 0.992·9-s + 0.694·10-s + 1.75·11-s − 0.0279·12-s − 0.179·13-s − 0.964·14-s − 0.0519·15-s − 1.21·16-s − 1.29·17-s − 1.14·18-s + 0.663·19-s + 0.196·20-s + 0.0720·21-s + 2.02·22-s + 0.555·23-s + 0.0667·24-s − 0.635·25-s − 0.206·26-s + 0.171·27-s − 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.475061766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475061766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 89 | \( 1 - T \) |
| good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 3 | \( 1 + 0.148T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 + 0.647T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 + 9.08T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 + 0.428T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 5.52T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 97 | \( 1 - 3.29T + 97T^{2} \) |
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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
−13.98801758949757004838048115145, −13.33585821725092292425384707523, −12.15171679301550827930629242671, −11.37822621401630300268161425648, −9.599053373796488450078227723737, −8.867916251429708211538924688531, −6.59052672984160179889265750991, −5.94986829848227808964312766630, −4.41029326705218714972973232598, −2.96274501696815526029123212601,
2.96274501696815526029123212601, 4.41029326705218714972973232598, 5.94986829848227808964312766630, 6.59052672984160179889265750991, 8.867916251429708211538924688531, 9.599053373796488450078227723737, 11.37822621401630300268161425648, 12.15171679301550827930629242671, 13.33585821725092292425384707523, 13.98801758949757004838048115145
