| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 7-s − 4·8-s + 3·9-s + 3·11-s − 6·12-s − 4·13-s + 2·14-s + 5·16-s + 2·17-s − 6·18-s + 19-s + 2·21-s − 6·22-s − 3·23-s + 8·24-s + 8·26-s − 4·27-s − 3·28-s + 6·29-s + 31-s − 6·32-s − 6·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s + 0.904·11-s − 1.73·12-s − 1.10·13-s + 0.534·14-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 0.229·19-s + 0.436·21-s − 1.27·22-s − 0.625·23-s + 1.63·24-s + 1.56·26-s − 0.769·27-s − 0.566·28-s + 1.11·29-s + 0.179·31-s − 1.06·32-s − 1.04·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6937450036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6937450036\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 120 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 132 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 160 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 154 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.160083140834387719937769567555, −8.759348817309569477299100728620, −8.292234543892338764905065940386, −8.047261086459208850021784057155, −7.37383425365714754395883232720, −7.33669877379976829630364242939, −6.72788400209651612664774457365, −6.62612090587766092980377353229, −6.04618233302466979237095192082, −5.85156969762441601096339750851, −5.27667623664469658032907857239, −4.87253417172046691818171132773, −4.33447282972658901595934647668, −3.95164368561694314257175396845, −3.08624244127148540310764745415, −2.99896598276881199310729623234, −1.99609609975692426308072017560, −1.77272738615959463739054834159, −0.876608763330026933228080071459, −0.50573883662229611494273106057,
0.50573883662229611494273106057, 0.876608763330026933228080071459, 1.77272738615959463739054834159, 1.99609609975692426308072017560, 2.99896598276881199310729623234, 3.08624244127148540310764745415, 3.95164368561694314257175396845, 4.33447282972658901595934647668, 4.87253417172046691818171132773, 5.27667623664469658032907857239, 5.85156969762441601096339750851, 6.04618233302466979237095192082, 6.62612090587766092980377353229, 6.72788400209651612664774457365, 7.33669877379976829630364242939, 7.37383425365714754395883232720, 8.047261086459208850021784057155, 8.292234543892338764905065940386, 8.759348817309569477299100728620, 9.160083140834387719937769567555
