| L(s) = 1 | + 2·2-s + 4-s + 2·5-s − 2·7-s + 4·10-s + 4·11-s − 4·13-s − 4·14-s + 16-s + 4·17-s + 2·20-s + 8·22-s + 4·23-s + 3·25-s − 8·26-s − 2·28-s + 16·29-s − 2·32-s + 8·34-s − 4·35-s − 12·37-s − 4·41-s − 8·43-s + 4·44-s + 8·46-s − 8·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 1.26·10-s + 1.20·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.447·20-s + 1.70·22-s + 0.834·23-s + 3/5·25-s − 1.56·26-s − 0.377·28-s + 2.97·29-s − 0.353·32-s + 1.37·34-s − 0.676·35-s − 1.97·37-s − 0.624·41-s − 1.21·43-s + 0.603·44-s + 1.17·46-s − 1.16·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.375747193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.375747193\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−11.98020686650259030586494199374, −11.88778227986212780790191364145, −11.05025484400275236394371234119, −10.24686923793886752589454070436, −10.03478918934206985266545904975, −9.914441309631567422457804698363, −8.912854602493676090650634141853, −8.857551043051971412794104378821, −8.143084229970569160877700535862, −7.23510411288316814009884652594, −6.73504098214906760824661374951, −6.63164496539452397848548466415, −5.77404950215446722334259996216, −5.35102125628084747039122442264, −4.78195276826434051493171186301, −4.49017964901229354309099325692, −3.37982943118690998095995547721, −3.34136922375041976081730783368, −2.32893067924470918397253396159, −1.25001904473718340900235882774,
1.25001904473718340900235882774, 2.32893067924470918397253396159, 3.34136922375041976081730783368, 3.37982943118690998095995547721, 4.49017964901229354309099325692, 4.78195276826434051493171186301, 5.35102125628084747039122442264, 5.77404950215446722334259996216, 6.63164496539452397848548466415, 6.73504098214906760824661374951, 7.23510411288316814009884652594, 8.143084229970569160877700535862, 8.857551043051971412794104378821, 8.912854602493676090650634141853, 9.914441309631567422457804698363, 10.03478918934206985266545904975, 10.24686923793886752589454070436, 11.05025484400275236394371234119, 11.88778227986212780790191364145, 11.98020686650259030586494199374
