| L(s) = 1 | − 2-s − 2·3-s − 2·5-s + 2·6-s + 8-s + 3·9-s + 2·10-s − 11-s − 4·13-s + 4·15-s − 16-s − 3·18-s − 2·19-s + 22-s − 2·24-s + 5·25-s + 4·26-s − 10·27-s + 12·29-s − 4·30-s + 4·31-s + 2·33-s − 2·37-s + 2·38-s + 8·39-s − 2·40-s − 16·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.353·8-s + 9-s + 0.632·10-s − 0.301·11-s − 1.10·13-s + 1.03·15-s − 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.213·22-s − 0.408·24-s + 25-s + 0.784·26-s − 1.92·27-s + 2.22·29-s − 0.730·30-s + 0.718·31-s + 0.348·33-s − 0.328·37-s + 0.324·38-s + 1.28·39-s − 0.316·40-s − 2.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5916642396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5916642396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−10.15162127456614335588445302723, −9.764401971721804061150702767473, −9.235566132530795811438169605879, −8.965511703508660890420321258391, −8.280772633380315411645403411256, −8.105255672730875594086235302454, −7.53001771158119094904028620260, −7.25822316400215267067921875143, −6.86083476069200721661233291799, −6.25133201944686583939766691088, −6.04830428888796741109664103242, −5.15069324339980276967040440648, −4.92199258564704079943606270906, −4.66727994084910739298855635168, −3.83169165403104546495645744047, −3.65604112789896656871695376719, −2.47323407831660167827795338654, −2.28446684875245789166645085976, −0.956570724107418190049854076104, −0.55390445976021708576146310196,
0.55390445976021708576146310196, 0.956570724107418190049854076104, 2.28446684875245789166645085976, 2.47323407831660167827795338654, 3.65604112789896656871695376719, 3.83169165403104546495645744047, 4.66727994084910739298855635168, 4.92199258564704079943606270906, 5.15069324339980276967040440648, 6.04830428888796741109664103242, 6.25133201944686583939766691088, 6.86083476069200721661233291799, 7.25822316400215267067921875143, 7.53001771158119094904028620260, 8.105255672730875594086235302454, 8.280772633380315411645403411256, 8.965511703508660890420321258391, 9.235566132530795811438169605879, 9.764401971721804061150702767473, 10.15162127456614335588445302723
