| L(s) = 1 | + 2·2-s + 4-s − 2·5-s − 4·7-s − 4·10-s + 4·11-s − 4·13-s − 8·14-s + 16-s − 4·19-s − 2·20-s + 8·22-s + 12·23-s + 3·25-s − 8·26-s − 4·28-s − 2·29-s − 4·31-s − 2·32-s + 8·35-s − 8·38-s + 12·41-s − 12·43-s + 4·44-s + 24·46-s + 12·47-s + 6·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 1.26·10-s + 1.20·11-s − 1.10·13-s − 2.13·14-s + 1/4·16-s − 0.917·19-s − 0.447·20-s + 1.70·22-s + 2.50·23-s + 3/5·25-s − 1.56·26-s − 0.755·28-s − 0.371·29-s − 0.718·31-s − 0.353·32-s + 1.35·35-s − 1.29·38-s + 1.87·41-s − 1.82·43-s + 0.603·44-s + 3.53·46-s + 1.75·47-s + 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.062375298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062375298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.631174227195902624716691172400, −9.569735796860283104887902752783, −8.935221445141604486704813754802, −8.886154431504759361613417113661, −8.228094159143384664934506081671, −7.58108352477538053422413409652, −7.13012654704060293700489516664, −7.02024890940965534265027683822, −6.51560178460189214488763299907, −6.14258378482332501032300726916, −5.42031063076244516808654166364, −5.24682186013868870229665504158, −4.50621496327616943337644256350, −4.38427934121461106876595539207, −3.68695708865252579033773835856, −3.61131050761625910666601968286, −2.90236261582425638504990454802, −2.55173832700983347070171173196, −1.47727387568019680231634409241, −0.48854540853707181561886723604,
0.48854540853707181561886723604, 1.47727387568019680231634409241, 2.55173832700983347070171173196, 2.90236261582425638504990454802, 3.61131050761625910666601968286, 3.68695708865252579033773835856, 4.38427934121461106876595539207, 4.50621496327616943337644256350, 5.24682186013868870229665504158, 5.42031063076244516808654166364, 6.14258378482332501032300726916, 6.51560178460189214488763299907, 7.02024890940965534265027683822, 7.13012654704060293700489516664, 7.58108352477538053422413409652, 8.228094159143384664934506081671, 8.886154431504759361613417113661, 8.935221445141604486704813754802, 9.569735796860283104887902752783, 9.631174227195902624716691172400
