| L(s) = 1 | + 2.28·3-s − 2.82·7-s + 2.23·9-s − 5.23·11-s − 4.03·13-s + 1.08·17-s − 19-s − 6.47·21-s + 7.40·23-s − 1.74·27-s + 4.47·29-s − 4·31-s − 11.9·33-s − 6.86·37-s − 9.23·39-s − 6·41-s − 8.48·43-s − 8.48·47-s + 1.00·49-s + 2.47·51-s + 6.86·53-s − 2.28·57-s − 10.4·59-s + 1.70·61-s − 6.32·63-s − 1.62·67-s + 16.9·69-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 1.06·7-s + 0.745·9-s − 1.57·11-s − 1.11·13-s + 0.262·17-s − 0.229·19-s − 1.41·21-s + 1.54·23-s − 0.336·27-s + 0.830·29-s − 0.718·31-s − 2.08·33-s − 1.12·37-s − 1.47·39-s − 0.937·41-s − 1.29·43-s − 1.23·47-s + 0.142·49-s + 0.346·51-s + 0.942·53-s − 0.303·57-s − 1.36·59-s + 0.218·61-s − 0.796·63-s − 0.197·67-s + 2.03·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 5.24T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 4.44T + 97T^{2} \) |
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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
−8.748580371067580704639475289959, −8.161011005377315314707426919718, −7.31899002408850242427015791774, −6.75373938335604072244306500622, −5.44673960962651922960646582450, −4.74208212326978026722927022325, −3.26110465758025879119597275130, −3.03408814518767089601946980012, −2.03770108810668503165609429346, 0,
2.03770108810668503165609429346, 3.03408814518767089601946980012, 3.26110465758025879119597275130, 4.74208212326978026722927022325, 5.44673960962651922960646582450, 6.75373938335604072244306500622, 7.31899002408850242427015791774, 8.161011005377315314707426919718, 8.748580371067580704639475289959
