L(s) = 1 | + 24·13-s − 30·17-s − 40·29-s + 24·37-s + 80·41-s + 49·49-s + 90·53-s + 22·61-s − 96·73-s − 160·89-s − 144·97-s + 40·101-s + 182·109-s + 30·113-s + ⋯ |
L(s) = 1 | + 1.84·13-s − 1.76·17-s − 1.37·29-s + 0.648·37-s + 1.95·41-s + 49-s + 1.69·53-s + 0.360·61-s − 1.31·73-s − 1.79·89-s − 1.48·97-s + 0.396·101-s + 1.66·109-s + 0.265·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.117962720\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.117962720\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 160 T + p^{2} T^{2} \) |
| 97 | \( 1 + 144 T + p^{2} T^{2} \) |
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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
−8.638656998437860564389916341246, −7.60928668600490316997195927834, −6.88606948817714453703301325783, −6.07042768656030463614812270609, −5.58007619112015283275121029220, −4.27622117263703739230920209889, −3.94444071200787772281139327649, −2.75984217204068231435226481324, −1.81419307973905666661919380508, −0.69388447839158712920590267613,
0.69388447839158712920590267613, 1.81419307973905666661919380508, 2.75984217204068231435226481324, 3.94444071200787772281139327649, 4.27622117263703739230920209889, 5.58007619112015283275121029220, 6.07042768656030463614812270609, 6.88606948817714453703301325783, 7.60928668600490316997195927834, 8.638656998437860564389916341246