L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 2.44·11-s + 3.44·13-s + 14-s + 16-s + 17-s + 0.449·19-s − 2.44·22-s − 3.44·23-s + 3.44·26-s + 28-s + 7.44·29-s + 3.44·31-s + 32-s + 34-s + 11.3·37-s + 0.449·38-s − 4.89·41-s − 5.89·43-s − 2.44·44-s − 3.44·46-s − 7.79·47-s + 49-s + 3.44·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s − 0.738·11-s + 0.956·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s + 0.103·19-s − 0.522·22-s − 0.719·23-s + 0.676·26-s + 0.188·28-s + 1.38·29-s + 0.619·31-s + 0.176·32-s + 0.171·34-s + 1.86·37-s + 0.0729·38-s − 0.765·41-s − 0.899·43-s − 0.369·44-s − 0.508·46-s − 1.13·47-s + 0.142·49-s + 0.478·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.718450859\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.718450859\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 5.89T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 0.550T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 1.10T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 0.550T + 71T^{2} \) |
| 73 | \( 1 + 7.79T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 + 1.10T + 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.84075557157435523448561589407, −6.81395576847602386781834500854, −6.27823175443810865365109843269, −5.61364988361176067337614526844, −4.88279557808545135657853000795, −4.30920079341088424504079191585, −3.44293651681996547965399257488, −2.75541601730156657205969424390, −1.86260391539857080119450498839, −0.846284165085445296909655847601,
0.846284165085445296909655847601, 1.86260391539857080119450498839, 2.75541601730156657205969424390, 3.44293651681996547965399257488, 4.30920079341088424504079191585, 4.88279557808545135657853000795, 5.61364988361176067337614526844, 6.27823175443810865365109843269, 6.81395576847602386781834500854, 7.84075557157435523448561589407