| L(s) = 1 | + 2.21·2-s + 2.91·4-s − 5-s − 3.75·7-s + 2.02·8-s − 2.21·10-s + 1.54·11-s + 4.15·13-s − 8.31·14-s − 1.33·16-s − 5.74·17-s + 7.03·19-s − 2.91·20-s + 3.41·22-s + 6.77·23-s + 25-s + 9.20·26-s − 10.9·28-s + 10.1·29-s + 0.0578·31-s − 7.01·32-s − 12.7·34-s + 3.75·35-s − 2.18·37-s + 15.5·38-s − 2.02·40-s + 8.82·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.45·4-s − 0.447·5-s − 1.41·7-s + 0.716·8-s − 0.700·10-s + 0.464·11-s + 1.15·13-s − 2.22·14-s − 0.334·16-s − 1.39·17-s + 1.61·19-s − 0.651·20-s + 0.728·22-s + 1.41·23-s + 0.200·25-s + 1.80·26-s − 2.06·28-s + 1.88·29-s + 0.0103·31-s − 1.24·32-s − 2.18·34-s + 0.634·35-s − 0.358·37-s + 2.52·38-s − 0.320·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.923669922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.923669922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 0.0578T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 3.49T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 8.83T + 79T^{2} \) |
| 83 | \( 1 - 9.09T + 83T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−8.548899108007283557911196992467, −7.25491243196850822738821483953, −6.62504763142322047285341611145, −6.32860391915425604926392204972, −5.35340137918502298661634930057, −4.61913186619548827444835039330, −3.74683221530736935911329436194, −3.28075390742662929475083161135, −2.55705563754935317431589997304, −0.904603493956570931623112412073,
0.904603493956570931623112412073, 2.55705563754935317431589997304, 3.28075390742662929475083161135, 3.74683221530736935911329436194, 4.61913186619548827444835039330, 5.35340137918502298661634930057, 6.32860391915425604926392204972, 6.62504763142322047285341611145, 7.25491243196850822738821483953, 8.548899108007283557911196992467
