| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 11-s + 6·13-s − 3·17-s + 5·19-s + 3·21-s − 2·23-s + 9·27-s − 5·29-s − 5·31-s + 3·33-s + 37-s + 18·39-s − 2·41-s + 12·43-s − 2·47-s − 6·49-s − 9·51-s + 13·53-s + 15·57-s − 2·59-s + 61-s + 6·63-s + 16·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 0.301·11-s + 1.66·13-s − 0.727·17-s + 1.14·19-s + 0.654·21-s − 0.417·23-s + 1.73·27-s − 0.928·29-s − 0.898·31-s + 0.522·33-s + 0.164·37-s + 2.88·39-s − 0.312·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s − 1.26·51-s + 1.78·53-s + 1.98·57-s − 0.260·59-s + 0.128·61-s + 0.755·63-s + 1.95·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.461690124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461690124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.581314018060219791053687549460, −7.67617895057151825096939663098, −7.26545715189733808033141127532, −6.25993733872492149184096201465, −5.39958993354332013291534258521, −4.16744073236887809039919802437, −3.78238631378783092682364329952, −2.96517269829137022097592262243, −2.00608371765600014153092741840, −1.23387887114154623856515596513,
1.23387887114154623856515596513, 2.00608371765600014153092741840, 2.96517269829137022097592262243, 3.78238631378783092682364329952, 4.16744073236887809039919802437, 5.39958993354332013291534258521, 6.25993733872492149184096201465, 7.26545715189733808033141127532, 7.67617895057151825096939663098, 8.581314018060219791053687549460
