| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s + 4·11-s + 5·16-s + 8·17-s + 8·19-s + 6·20-s + 8·22-s − 12·23-s + 3·25-s − 8·29-s + 12·31-s + 6·32-s + 16·34-s + 8·37-s + 16·38-s + 8·40-s + 4·41-s + 4·43-s + 12·44-s − 24·46-s + 4·47-s + 6·50-s − 12·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 1.20·11-s + 5/4·16-s + 1.94·17-s + 1.83·19-s + 1.34·20-s + 1.70·22-s − 2.50·23-s + 3/5·25-s − 1.48·29-s + 2.15·31-s + 1.06·32-s + 2.74·34-s + 1.31·37-s + 2.59·38-s + 1.26·40-s + 0.624·41-s + 0.609·43-s + 1.80·44-s − 3.53·46-s + 0.583·47-s + 0.848·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.92547986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.92547986\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.258390320174948501391142978796, −7.970041577216843989306690371804, −7.67650417120577189564121916188, −7.61638646735912957589467093045, −6.87357322509909134828516681925, −6.53716949014371413480786572832, −6.13218875671852155212025933019, −5.97648325036665893431123155007, −5.63076604613519193332043577231, −5.31172770008620258896694634626, −4.81916269277032597488056799899, −4.45949561848318821920705205574, −3.82840165900630956534830222101, −3.76370318986589149415218509904, −3.24967061873248773931801190132, −2.78114868448936624469133823256, −2.32154391306376787407835723753, −1.80696067237084208739112015612, −1.21455973092262750591897652752, −0.912033113218131558545210907976,
0.912033113218131558545210907976, 1.21455973092262750591897652752, 1.80696067237084208739112015612, 2.32154391306376787407835723753, 2.78114868448936624469133823256, 3.24967061873248773931801190132, 3.76370318986589149415218509904, 3.82840165900630956534830222101, 4.45949561848318821920705205574, 4.81916269277032597488056799899, 5.31172770008620258896694634626, 5.63076604613519193332043577231, 5.97648325036665893431123155007, 6.13218875671852155212025933019, 6.53716949014371413480786572832, 6.87357322509909134828516681925, 7.61638646735912957589467093045, 7.67650417120577189564121916188, 7.970041577216843989306690371804, 8.258390320174948501391142978796
