| L(s) = 1 | + 4·2-s − 3-s + 12·4-s − 4·6-s + 27·7-s + 32·8-s + 25·9-s − 52·11-s − 12·12-s + 17·13-s + 108·14-s + 80·16-s − 75·17-s + 100·18-s + 38·19-s − 27·21-s − 208·22-s + 203·23-s − 32·24-s + 68·26-s − 76·27-s + 324·28-s + 183·29-s − 18·31-s + 192·32-s + 52·33-s − 300·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s + 1.45·7-s + 1.41·8-s + 0.925·9-s − 1.42·11-s − 0.288·12-s + 0.362·13-s + 2.06·14-s + 5/4·16-s − 1.07·17-s + 1.30·18-s + 0.458·19-s − 0.280·21-s − 2.01·22-s + 1.84·23-s − 0.272·24-s + 0.512·26-s − 0.541·27-s + 2.18·28-s + 1.17·29-s − 0.104·31-s + 1.06·32-s + 0.274·33-s − 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.07667592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.07667592\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T - 8 p T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 27 T + 790 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 52 T + 2086 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17 T + 3762 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 75 T + 10528 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 203 T + 30802 T^{2} - 203 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 183 T + 50812 T^{2} - 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 18 T + 6766 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 78 T + 64954 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 390 T + p^{3} T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 338 T + 187262 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 24 T + 162718 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 77950 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 687 T + 370294 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1222 T + 811946 T^{2} + 1222 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 43 T + 250724 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1500 T + 1176910 T^{2} - 1500 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 983 T + 900588 T^{2} - 983 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 750 T + 1126390 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1010 T + 819862 T^{2} + 1010 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 164 T + 810694 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 954 T + 1301362 T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.970142502673178543521706429951, −9.568975185464477067828508893448, −8.962102782033766914122155121963, −8.515577894766183633104227004063, −7.985851859228781972512507727804, −7.50089586082315143029449144745, −7.48062792106666494278073985435, −6.83372359829170419262197063014, −6.15851405718921442167246758724, −6.11911590948591655652302054274, −5.13464245421887092332178388995, −4.95956734496589551197580896164, −4.77215001675389421853622134617, −4.33082836982738947756603490264, −3.60713698357389011860136956416, −3.05531948206261635909907604664, −2.50328471731616364515862116782, −1.94780221624124080936063720390, −1.35088218372709137864397903339, −0.68115266220540391884769402820,
0.68115266220540391884769402820, 1.35088218372709137864397903339, 1.94780221624124080936063720390, 2.50328471731616364515862116782, 3.05531948206261635909907604664, 3.60713698357389011860136956416, 4.33082836982738947756603490264, 4.77215001675389421853622134617, 4.95956734496589551197580896164, 5.13464245421887092332178388995, 6.11911590948591655652302054274, 6.15851405718921442167246758724, 6.83372359829170419262197063014, 7.48062792106666494278073985435, 7.50089586082315143029449144745, 7.985851859228781972512507727804, 8.515577894766183633104227004063, 8.962102782033766914122155121963, 9.568975185464477067828508893448, 9.970142502673178543521706429951
