| L(s) = 1 | + 2·7-s + 14·13-s − 14·19-s + 34·31-s + 32·37-s + 110·43-s − 95·49-s + 130·61-s + 98·67-s + 176·73-s − 80·79-s + 28·91-s − 82·97-s + 164·103-s − 98·109-s + 224·121-s + 127-s + 131-s − 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + ⋯ |
| L(s) = 1 | + 2/7·7-s + 1.07·13-s − 0.736·19-s + 1.09·31-s + 0.864·37-s + 2.55·43-s − 1.93·49-s + 2.13·61-s + 1.46·67-s + 2.41·73-s − 1.01·79-s + 4/13·91-s − 0.845·97-s + 1.59·103-s − 0.899·109-s + 1.85·121-s + 0.00787·127-s + 0.00763·131-s − 0.210·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.108890492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.108890492\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 176 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 800 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 55 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2240 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1582 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3920 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 88 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10864 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{2} \) |
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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
−9.907467649832000497912187578882, −9.833376969806707830951283907990, −9.340488369545554888684398464750, −8.766715718605769084927511130570, −8.314522074295510227305328740979, −8.241304989791121879502021766723, −7.64449076527837287694459691226, −7.16023420339230932141109147780, −6.53081296600648337006074799463, −6.37186327173828312955884966468, −5.77368050017941388068193082583, −5.41254394100573927992823592097, −4.65297367975343320918081502541, −4.40647406764850872835359828941, −3.75487489311951140460769370037, −3.36124969219007441285993405610, −2.49226740001684189182400988810, −2.15300780310458775800436869368, −1.18927165746010888507052262645, −0.65379394620780875393636780049,
0.65379394620780875393636780049, 1.18927165746010888507052262645, 2.15300780310458775800436869368, 2.49226740001684189182400988810, 3.36124969219007441285993405610, 3.75487489311951140460769370037, 4.40647406764850872835359828941, 4.65297367975343320918081502541, 5.41254394100573927992823592097, 5.77368050017941388068193082583, 6.37186327173828312955884966468, 6.53081296600648337006074799463, 7.16023420339230932141109147780, 7.64449076527837287694459691226, 8.241304989791121879502021766723, 8.314522074295510227305328740979, 8.766715718605769084927511130570, 9.340488369545554888684398464750, 9.833376969806707830951283907990, 9.907467649832000497912187578882
