| L(s) = 1 | − 3-s + 4·7-s + 3·9-s − 3·11-s + 12·13-s − 5·17-s − 19-s − 4·21-s − 7·23-s − 8·27-s + 4·29-s + 5·31-s + 3·33-s + 3·37-s − 12·39-s − 4·41-s + 8·43-s + 5·47-s + 9·49-s + 5·51-s − 53-s + 57-s − 15·59-s + 5·61-s + 12·63-s − 9·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 9-s − 0.904·11-s + 3.32·13-s − 1.21·17-s − 0.229·19-s − 0.872·21-s − 1.45·23-s − 1.53·27-s + 0.742·29-s + 0.898·31-s + 0.522·33-s + 0.493·37-s − 1.92·39-s − 0.624·41-s + 1.21·43-s + 0.729·47-s + 9/7·49-s + 0.700·51-s − 0.137·53-s + 0.132·57-s − 1.95·59-s + 0.640·61-s + 1.51·63-s − 1.09·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536313560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536313560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.987170380933182292777934332797, −9.248931491442342479376431271500, −8.814930063453752647490213279045, −8.404044813395472822934528611437, −8.343033125254692212301780658008, −7.72654194938336075444074332158, −7.52233515812876840574795672031, −6.84709508040139028640097800348, −6.27294652942890683990388644762, −6.11773983284387323927250460273, −5.71448049899111817187160039404, −5.24937498851628406251161061580, −4.51101996962661472736605195982, −4.21887654080069622407440767733, −4.10629585649288641949660291513, −3.31496916892761515790094972824, −2.56982473408329351555263787734, −1.70876384338215643278427297682, −1.58333758748498670802724269650, −0.71694095683985791390190510262,
0.71694095683985791390190510262, 1.58333758748498670802724269650, 1.70876384338215643278427297682, 2.56982473408329351555263787734, 3.31496916892761515790094972824, 4.10629585649288641949660291513, 4.21887654080069622407440767733, 4.51101996962661472736605195982, 5.24937498851628406251161061580, 5.71448049899111817187160039404, 6.11773983284387323927250460273, 6.27294652942890683990388644762, 6.84709508040139028640097800348, 7.52233515812876840574795672031, 7.72654194938336075444074332158, 8.343033125254692212301780658008, 8.404044813395472822934528611437, 8.814930063453752647490213279045, 9.248931491442342479376431271500, 9.987170380933182292777934332797
