| L(s) = 1 | − 2·2-s + 6·4-s − 26·7-s − 32·8-s − 28·11-s − 18·13-s + 52·14-s + 44·16-s − 68·17-s + 6·19-s + 56·22-s + 132·23-s + 36·26-s − 156·28-s − 92·29-s + 122·31-s − 120·32-s + 136·34-s + 284·37-s − 12·38-s − 392·41-s − 690·43-s − 168·44-s − 264·46-s − 620·47-s + 125·49-s − 108·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3/4·4-s − 1.40·7-s − 1.41·8-s − 0.767·11-s − 0.384·13-s + 0.992·14-s + 0.687·16-s − 0.970·17-s + 0.0724·19-s + 0.542·22-s + 1.19·23-s + 0.271·26-s − 1.05·28-s − 0.589·29-s + 0.706·31-s − 0.662·32-s + 0.685·34-s + 1.26·37-s − 0.0512·38-s − 1.49·41-s − 2.44·43-s − 0.575·44-s − 0.846·46-s − 1.92·47-s + 0.364·49-s − 0.288·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T - p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 26 T + 551 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 28 T + 2554 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 18 T - 389 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 p T + 3382 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 13423 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 132 T + 25954 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 92 T + 35998 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 122 T + 48407 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 284 T + 110526 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 392 T + 124882 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 690 T + 277735 T^{2} + 690 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 620 T + 284290 T^{2} + 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 p T + 462634 T^{2} + 16 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 124 T + 336778 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 750 T + 535003 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 358 T + 443567 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 824 T + 877966 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 108 T + 779734 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 880 T + 693278 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 156 T + 449242 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 864 T + 1202578 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 521 T + p^{3} T^{2} )^{2} \) |
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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
−11.55021960122252411399523205809, −11.12478540148072615824046553444, −10.48855040014758050701933772578, −9.937044750743145506409912959274, −9.548783103144742172932981125637, −9.365353954956778813155799824565, −8.428982533696333969387209869995, −8.367209735263747923382268320383, −7.52759664358794988735516251208, −6.76228193913428075692672840871, −6.60261573530341376257426617060, −6.21528123781056840485155092775, −5.24375285467712430643393530830, −4.83743015699844166885344310706, −3.67197270933323698391309349481, −2.96396610941625342550156613737, −2.71263975134744512745668581964, −1.59458930619225039200337614534, 0, 0,
1.59458930619225039200337614534, 2.71263975134744512745668581964, 2.96396610941625342550156613737, 3.67197270933323698391309349481, 4.83743015699844166885344310706, 5.24375285467712430643393530830, 6.21528123781056840485155092775, 6.60261573530341376257426617060, 6.76228193913428075692672840871, 7.52759664358794988735516251208, 8.367209735263747923382268320383, 8.428982533696333969387209869995, 9.365353954956778813155799824565, 9.548783103144742172932981125637, 9.937044750743145506409912959274, 10.48855040014758050701933772578, 11.12478540148072615824046553444, 11.55021960122252411399523205809
