| L(s) = 1 | − 2·2-s − 3-s + 2·6-s + 20·7-s + 8·8-s + 27·9-s − 35·11-s − 132·13-s − 40·14-s − 16·16-s + 59·17-s − 54·18-s − 137·19-s − 20·21-s + 70·22-s − 7·23-s − 8·24-s + 264·26-s − 80·27-s + 212·29-s − 75·31-s + 35·33-s − 118·34-s + 11·37-s + 274·38-s + 132·39-s − 996·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 0.136·6-s + 1.07·7-s + 0.353·8-s + 9-s − 0.959·11-s − 2.81·13-s − 0.763·14-s − 1/4·16-s + 0.841·17-s − 0.707·18-s − 1.65·19-s − 0.207·21-s + 0.678·22-s − 0.0634·23-s − 0.0680·24-s + 1.99·26-s − 0.570·27-s + 1.35·29-s − 0.434·31-s + 0.184·33-s − 0.595·34-s + 0.0488·37-s + 1.16·38-s + 0.541·39-s − 3.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1476574765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1476574765\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 20 T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 35 T - 106 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 59 T - 1432 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 137 T + 11910 T^{2} + 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T - 12118 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 75 T - 24166 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 11 T - 50532 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 498 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 260 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 171 T - 74582 T^{2} + 171 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 417 T + 25012 T^{2} + 417 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 17 T - 205090 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 439 T - 108042 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 784 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 295 T - 301992 T^{2} - 295 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 932 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
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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
−11.44499634760613249600918936028, −10.41816610053329578691005083906, −10.27313561511752907770263618492, −10.00372359237019322727147659437, −9.780593995027672180449217878621, −8.782262332110583015717920623212, −8.530726450427554341367043069230, −7.84131566100644516096246478486, −7.77771793735576789885569481978, −6.93693587096006716564788222343, −6.92218878783427357446642331818, −5.96494565749609450939243639067, −4.96023506311919722308434974509, −4.81390782737162006274418269089, −4.77837759845928790766213766283, −3.60052821760592070351233374361, −2.75248657316652977619590646714, −1.92961064648979968451739817522, −1.58885509511434552720671401737, −0.14923568357506035820431349329,
0.14923568357506035820431349329, 1.58885509511434552720671401737, 1.92961064648979968451739817522, 2.75248657316652977619590646714, 3.60052821760592070351233374361, 4.77837759845928790766213766283, 4.81390782737162006274418269089, 4.96023506311919722308434974509, 5.96494565749609450939243639067, 6.92218878783427357446642331818, 6.93693587096006716564788222343, 7.77771793735576789885569481978, 7.84131566100644516096246478486, 8.530726450427554341367043069230, 8.782262332110583015717920623212, 9.780593995027672180449217878621, 10.00372359237019322727147659437, 10.27313561511752907770263618492, 10.41816610053329578691005083906, 11.44499634760613249600918936028
