L(s) = 1 | − 6·5-s − 7·7-s − 12·11-s − 82·13-s + 30·17-s − 68·19-s + 216·23-s − 89·25-s − 246·29-s + 112·31-s + 42·35-s + 110·37-s + 246·41-s + 172·43-s + 192·47-s + 49·49-s − 558·53-s + 72·55-s + 540·59-s + 110·61-s + 492·65-s − 140·67-s − 840·71-s − 550·73-s + 84·77-s + 208·79-s + 516·83-s + ⋯ |
L(s) = 1 | − 0.536·5-s − 0.377·7-s − 0.328·11-s − 1.74·13-s + 0.428·17-s − 0.821·19-s + 1.95·23-s − 0.711·25-s − 1.57·29-s + 0.648·31-s + 0.202·35-s + 0.488·37-s + 0.937·41-s + 0.609·43-s + 0.595·47-s + 1/7·49-s − 1.44·53-s + 0.176·55-s + 1.19·59-s + 0.230·61-s + 0.938·65-s − 0.255·67-s − 1.40·71-s − 0.881·73-s + 0.124·77-s + 0.296·79-s + 0.682·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.122983011\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122983011\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 216 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 140 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 550 T + p^{3} T^{2} \) |
| 79 | \( 1 - 208 T + p^{3} T^{2} \) |
| 83 | \( 1 - 516 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1586 T + p^{3} T^{2} \) |
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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
−9.555306773507720860577488870593, −8.867913280505902025326905783237, −7.59382569275539513688835172809, −7.38355710508562751144968135554, −6.18426148452999220828357010451, −5.15539095999123937951856941625, −4.33768494447444029632537673545, −3.18293892443071277236759996534, −2.23326118303487741740677473394, −0.53933110866247915680393449368,
0.53933110866247915680393449368, 2.23326118303487741740677473394, 3.18293892443071277236759996534, 4.33768494447444029632537673545, 5.15539095999123937951856941625, 6.18426148452999220828357010451, 7.38355710508562751144968135554, 7.59382569275539513688835172809, 8.867913280505902025326905783237, 9.555306773507720860577488870593