| L(s) = 1 | − 4·2-s − 9·3-s − 66·5-s + 36·6-s + 64·8-s + 264·10-s + 60·11-s + 1.31e3·13-s + 594·15-s − 256·16-s − 414·17-s + 956·19-s − 240·22-s − 600·23-s − 576·24-s + 3.12e3·25-s − 5.26e3·26-s + 729·27-s + 1.11e4·29-s − 2.37e3·30-s − 3.59e3·31-s − 540·33-s + 1.65e3·34-s + 8.45e3·37-s − 3.82e3·38-s − 1.18e4·39-s − 4.22e3·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1.18·5-s + 0.408·6-s + 0.353·8-s + 0.834·10-s + 0.149·11-s + 2.15·13-s + 0.681·15-s − 1/4·16-s − 0.347·17-s + 0.607·19-s − 0.105·22-s − 0.236·23-s − 0.204·24-s + 25-s − 1.52·26-s + 0.192·27-s + 2.46·29-s − 0.481·30-s − 0.671·31-s − 0.0863·33-s + 0.245·34-s + 1.01·37-s − 0.429·38-s − 1.24·39-s − 0.417·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8173334992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8173334992\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 66 T + 1231 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 60 T - 157451 T^{2} - 60 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 658 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 414 T - 1248461 T^{2} + 414 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 956 T - 1562163 T^{2} - 956 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 600 T - 6076343 T^{2} + 600 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5574 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3592 T - 15726687 T^{2} + 3592 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8458 T + 2193807 T^{2} - 8458 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 19194 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13316 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 19680 T + 157957393 T^{2} + 19680 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31266 T + 559367263 T^{2} - 31266 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 26340 T - 21128699 T^{2} - 26340 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 31090 T + 121991799 T^{2} + 31090 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 16804 T - 1067750691 T^{2} - 16804 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 25558 T - 1419860229 T^{2} + 25558 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 74408 T + 2459494065 T^{2} + 74408 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6468 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 32742 T - 4512020885 T^{2} + 32742 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 166082 T + p^{5} T^{2} )^{2} \) |
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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
−11.16405690785750962517957729520, −10.59666994623695620131299422592, −10.51185963827847038168948863488, −9.731704632076195621139454986724, −9.241316422756062759757480390361, −8.544790368079452624790885755570, −8.291098781389758607023202823617, −8.187325767977830632337476066303, −7.28652246979139996360830080716, −6.61929038410843121444332952987, −6.61750436039157874416055478279, −5.64158934229286571932637687758, −5.26319697609641260632932560471, −4.26530410902136749831566192152, −4.16073735690507415198346542558, −3.33930705836991288869032007897, −2.77355139327966991472389943108, −1.46689882851051516437205575695, −1.07341417662103448072030332220, −0.35789569649946744305125083339,
0.35789569649946744305125083339, 1.07341417662103448072030332220, 1.46689882851051516437205575695, 2.77355139327966991472389943108, 3.33930705836991288869032007897, 4.16073735690507415198346542558, 4.26530410902136749831566192152, 5.26319697609641260632932560471, 5.64158934229286571932637687758, 6.61750436039157874416055478279, 6.61929038410843121444332952987, 7.28652246979139996360830080716, 8.187325767977830632337476066303, 8.291098781389758607023202823617, 8.544790368079452624790885755570, 9.241316422756062759757480390361, 9.731704632076195621139454986724, 10.51185963827847038168948863488, 10.59666994623695620131299422592, 11.16405690785750962517957729520
