| L(s) = 1 | − 2·5-s − 7-s + 3·17-s + 2·19-s + 23-s − 25-s + 10·29-s − 5·31-s + 2·35-s + 2·37-s − 10·41-s − 43-s − 10·47-s + 49-s + 9·53-s + 9·59-s − 61-s + 9·67-s + 9·71-s + 4·73-s + 4·79-s + 83-s − 6·85-s − 15·89-s − 4·95-s − 12·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.727·17-s + 0.458·19-s + 0.208·23-s − 1/5·25-s + 1.85·29-s − 0.898·31-s + 0.338·35-s + 0.328·37-s − 1.56·41-s − 0.152·43-s − 1.45·47-s + 1/7·49-s + 1.23·53-s + 1.17·59-s − 0.128·61-s + 1.09·67-s + 1.06·71-s + 0.468·73-s + 0.450·79-s + 0.109·83-s − 0.650·85-s − 1.58·89-s − 0.410·95-s − 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577609566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577609566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.93774898114155, −13.42685286882322, −12.90249244235131, −12.34429777860209, −11.94910891866009, −11.55882403477768, −11.03642919820964, −10.40812635012006, −9.858467441530839, −9.618346602860770, −8.738582359281106, −8.330390703373521, −7.931502249600178, −7.321936650734622, −6.726353984226174, −6.455134668881581, −5.410567394672228, −5.265973934173616, −4.432253793832020, −3.842276900912773, −3.362465881972872, −2.844392654936166, −2.013962854102459, −1.142234434651273, −0.4515136456024868,
0.4515136456024868, 1.142234434651273, 2.013962854102459, 2.844392654936166, 3.362465881972872, 3.842276900912773, 4.432253793832020, 5.265973934173616, 5.410567394672228, 6.455134668881581, 6.726353984226174, 7.321936650734622, 7.931502249600178, 8.330390703373521, 8.738582359281106, 9.618346602860770, 9.858467441530839, 10.40812635012006, 11.03642919820964, 11.55882403477768, 11.94910891866009, 12.34429777860209, 12.90249244235131, 13.42685286882322, 13.93774898114155
