| L(s) = 1 | + 2-s − 8·5-s + 4·7-s − 8-s − 8·10-s − 4·11-s − 6·13-s + 4·14-s − 16-s + 2·17-s + 4·19-s − 4·22-s + 2·23-s + 38·25-s − 6·26-s + 4·29-s + 9·31-s + 2·34-s − 32·35-s − 8·37-s + 4·38-s + 8·40-s + 3·41-s − 2·43-s + 2·46-s − 9·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3.57·5-s + 1.51·7-s − 0.353·8-s − 2.52·10-s − 1.20·11-s − 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s + 0.417·23-s + 38/5·25-s − 1.17·26-s + 0.742·29-s + 1.61·31-s + 0.342·34-s − 5.40·35-s − 1.31·37-s + 0.648·38-s + 1.26·40-s + 0.468·41-s − 0.304·43-s + 0.294·46-s − 1.31·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6636507068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6636507068\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 113 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.20722375489288034585671516326, −9.608894106938001844731421185868, −9.070489920542411952601940459959, −8.251280628711767495856438442276, −8.221340299743421962187666103657, −8.102292066655922722400049252631, −7.50683348517793178960040618664, −7.42797481770432500044546182042, −6.90838370749308142028053775992, −6.40946812000155733098434213947, −5.21160667594789698229733972020, −5.07199259717945479110504043875, −4.83974780873966063684757016842, −4.52187505609357207725031138406, −3.88091180435206513628033822569, −3.55549158929830003379818731648, −2.83260179312313513232665017106, −2.69922148360612209446815807452, −1.26199554576020888487092176013, −0.35421968956502383157987901435,
0.35421968956502383157987901435, 1.26199554576020888487092176013, 2.69922148360612209446815807452, 2.83260179312313513232665017106, 3.55549158929830003379818731648, 3.88091180435206513628033822569, 4.52187505609357207725031138406, 4.83974780873966063684757016842, 5.07199259717945479110504043875, 5.21160667594789698229733972020, 6.40946812000155733098434213947, 6.90838370749308142028053775992, 7.42797481770432500044546182042, 7.50683348517793178960040618664, 8.102292066655922722400049252631, 8.221340299743421962187666103657, 8.251280628711767495856438442276, 9.070489920542411952601940459959, 9.608894106938001844731421185868, 10.20722375489288034585671516326
