| L(s) = 1 | − 8·3-s + 16·7-s + 37·9-s − 40·11-s − 50·13-s + 30·17-s + 40·19-s − 128·21-s + 48·23-s − 80·27-s + 34·29-s − 320·31-s + 320·33-s + 310·37-s + 400·39-s + 410·41-s − 152·43-s − 416·47-s − 87·49-s − 240·51-s − 410·53-s − 320·57-s − 200·59-s − 30·61-s + 592·63-s − 776·67-s − 384·69-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.863·7-s + 1.37·9-s − 1.09·11-s − 1.06·13-s + 0.428·17-s + 0.482·19-s − 1.33·21-s + 0.435·23-s − 0.570·27-s + 0.217·29-s − 1.85·31-s + 1.68·33-s + 1.37·37-s + 1.64·39-s + 1.56·41-s − 0.539·43-s − 1.29·47-s − 0.253·49-s − 0.658·51-s − 1.06·53-s − 0.743·57-s − 0.441·59-s − 0.0629·61-s + 1.18·63-s − 1.41·67-s − 0.669·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8083162300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8083162300\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 416 T + p^{3} T^{2} \) |
| 53 | \( 1 + 410 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−9.246057983553326163200671499598, −7.85307592800567424949871789418, −7.55419681364820494835359746952, −6.50176061984022841871306569955, −5.54358639289057795733329563901, −5.10280849466233146169527442782, −4.43656617653808281793710472000, −2.92455275558375989384975319898, −1.66086619973863982051246998957, −0.47444034053957629981648384660,
0.47444034053957629981648384660, 1.66086619973863982051246998957, 2.92455275558375989384975319898, 4.43656617653808281793710472000, 5.10280849466233146169527442782, 5.54358639289057795733329563901, 6.50176061984022841871306569955, 7.55419681364820494835359746952, 7.85307592800567424949871789418, 9.246057983553326163200671499598
