| L(s) = 1 | + 39·4-s + 450·9-s + 1.58e3·11-s + 497·16-s − 5.41e3·19-s + 1.00e4·29-s − 7.26e3·31-s + 1.75e4·36-s − 588·41-s + 6.19e4·44-s − 2.59e4·49-s + 6.01e4·59-s − 1.16e4·61-s − 2.05e4·64-s + 9.46e3·71-s − 2.11e5·76-s + 7.96e4·79-s + 1.43e5·81-s − 1.59e4·89-s + 7.14e5·99-s − 4.60e4·101-s + 3.91e5·109-s + 3.92e5·116-s + 1.56e6·121-s − 2.83e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.21·4-s + 1.85·9-s + 3.95·11-s + 0.485·16-s − 3.43·19-s + 2.22·29-s − 1.35·31-s + 2.25·36-s − 0.0546·41-s + 4.82·44-s − 1.54·49-s + 2.24·59-s − 0.399·61-s − 0.627·64-s + 0.222·71-s − 4.19·76-s + 1.43·79-s + 2.42·81-s − 0.213·89-s + 7.32·99-s − 0.448·101-s + 3.15·109-s + 2.71·116-s + 9.74·121-s − 1.65·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.725242696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.725242696\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 39 T^{2} + p^{10} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 50 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 25922 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 794 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 486558 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2706 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12379882 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5038 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3634 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 88872550 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 294 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 235982962 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 449569614 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 835999110 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 30066 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5806 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2545596118 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4734 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3930229550 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 39804 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6132847110 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 7970 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11082878014 T^{2} + p^{10} T^{4} \) |
| 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
−10.90704668785456674378082875608, −10.75748950898616657756193283329, −10.00279361484659106920487348075, −9.730524335571496620734177231365, −9.136078248987469149855786524773, −8.586648962569661670380884630007, −8.432270618504920561227273063467, −7.34318604133491691675136190807, −6.94583284412465461775232356622, −6.61946179173071080713170516950, −6.36234121000902572909798993348, −6.14619396535552541230593474087, −4.59627163949371334436113529157, −4.48012680351800343502577175334, −3.84284579471849154955712984821, −3.55006360939905682187576633600, −2.21792017983873206676462031995, −1.84587647708430760271383295688, −1.40034635527023718921485214108, −0.74977133569411520478731274601,
0.74977133569411520478731274601, 1.40034635527023718921485214108, 1.84587647708430760271383295688, 2.21792017983873206676462031995, 3.55006360939905682187576633600, 3.84284579471849154955712984821, 4.48012680351800343502577175334, 4.59627163949371334436113529157, 6.14619396535552541230593474087, 6.36234121000902572909798993348, 6.61946179173071080713170516950, 6.94583284412465461775232356622, 7.34318604133491691675136190807, 8.432270618504920561227273063467, 8.586648962569661670380884630007, 9.136078248987469149855786524773, 9.730524335571496620734177231365, 10.00279361484659106920487348075, 10.75748950898616657756193283329, 10.90704668785456674378082875608
