| L(s) = 1 | − 2·2-s − 4-s + 8·8-s + 9-s + 8·11-s − 7·16-s − 2·18-s − 16·22-s + 16·23-s + 25-s − 4·29-s − 14·32-s − 36-s + 12·37-s − 8·43-s − 8·44-s − 32·46-s − 7·49-s − 2·50-s + 12·53-s + 8·58-s + 35·64-s − 8·67-s + 8·72-s − 24·74-s + 32·79-s + 81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1/3·9-s + 2.41·11-s − 7/4·16-s − 0.471·18-s − 3.41·22-s + 3.33·23-s + 1/5·25-s − 0.742·29-s − 2.47·32-s − 1/6·36-s + 1.97·37-s − 1.21·43-s − 1.20·44-s − 4.71·46-s − 49-s − 0.282·50-s + 1.64·53-s + 1.05·58-s + 35/8·64-s − 0.977·67-s + 0.942·72-s − 2.78·74-s + 3.60·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120692382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120692382\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.66513698740624617246413812906, −7.66093767427808108694408525270, −7.08802589844853263958022569329, −6.54785831295780264103860348787, −6.48654966834348628333399350819, −5.57900643282056552246174341208, −4.88823595164803979266926983864, −4.87469251592617747564502770211, −4.27586819251074296637919964352, −3.61843511693942583930939957195, −3.56799192480000330057521215275, −2.49981243316321035709858596980, −1.53827946418072491307494025034, −1.16776210284876177743507098510, −0.73264725064818620387577910362,
0.73264725064818620387577910362, 1.16776210284876177743507098510, 1.53827946418072491307494025034, 2.49981243316321035709858596980, 3.56799192480000330057521215275, 3.61843511693942583930939957195, 4.27586819251074296637919964352, 4.87469251592617747564502770211, 4.88823595164803979266926983864, 5.57900643282056552246174341208, 6.48654966834348628333399350819, 6.54785831295780264103860348787, 7.08802589844853263958022569329, 7.66093767427808108694408525270, 7.66513698740624617246413812906
