| L(s) = 1 | − 2-s − 7·4-s − 5·5-s + 15·8-s + 5·10-s + 44·11-s − 6·13-s + 41·16-s − 24·17-s + 114·19-s + 35·20-s − 44·22-s + 52·23-s + 25·25-s + 6·26-s − 146·29-s + 276·31-s − 161·32-s + 24·34-s − 210·37-s − 114·38-s − 75·40-s + 444·41-s + 492·43-s − 308·44-s − 52·46-s − 612·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.447·5-s + 0.662·8-s + 0.158·10-s + 1.20·11-s − 0.128·13-s + 0.640·16-s − 0.342·17-s + 1.37·19-s + 0.391·20-s − 0.426·22-s + 0.471·23-s + 1/5·25-s + 0.0452·26-s − 0.934·29-s + 1.59·31-s − 0.889·32-s + 0.121·34-s − 0.933·37-s − 0.486·38-s − 0.296·40-s + 1.69·41-s + 1.74·43-s − 1.05·44-s − 0.166·46-s − 1.89·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.418431155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418431155\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 146 T + p^{3} T^{2} \) |
| 31 | \( 1 - 276 T + p^{3} T^{2} \) |
| 37 | \( 1 + 210 T + p^{3} T^{2} \) |
| 41 | \( 1 - 444 T + p^{3} T^{2} \) |
| 43 | \( 1 - 492 T + p^{3} T^{2} \) |
| 47 | \( 1 + 612 T + p^{3} T^{2} \) |
| 53 | \( 1 + 50 T + p^{3} T^{2} \) |
| 59 | \( 1 - 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 450 T + p^{3} T^{2} \) |
| 67 | \( 1 + 668 T + p^{3} T^{2} \) |
| 71 | \( 1 - 308 T + p^{3} T^{2} \) |
| 73 | \( 1 + 12 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 + 966 T + p^{3} T^{2} \) |
| 89 | \( 1 + 408 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1200 T + p^{3} T^{2} \) |
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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
−8.887582016180446485661869864926, −7.937694504644709292347766949509, −7.36801119015700849546245765306, −6.43406208348688481997916817866, −5.44607373075682657394236716836, −4.56486391154177455406456970960, −3.91786864302479145939291615917, −2.99203464486258563179003441174, −1.44828322128822840539018821088, −0.63294142068202661634291321670,
0.63294142068202661634291321670, 1.44828322128822840539018821088, 2.99203464486258563179003441174, 3.91786864302479145939291615917, 4.56486391154177455406456970960, 5.44607373075682657394236716836, 6.43406208348688481997916817866, 7.36801119015700849546245765306, 7.937694504644709292347766949509, 8.887582016180446485661869864926
