| L(s) = 1 | − 4·9-s − 4·11-s − 4·13-s − 4·17-s + 2·19-s + 8·23-s + 4·29-s − 12·37-s + 4·41-s − 6·49-s − 12·53-s + 8·61-s − 16·67-s − 8·71-s + 4·73-s + 24·79-s + 7·81-s + 16·83-s − 12·89-s − 4·97-s + 16·99-s + 24·101-s + 8·103-s − 4·109-s + 4·113-s + 16·117-s − 2·121-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 1.20·11-s − 1.10·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s + 0.742·29-s − 1.97·37-s + 0.624·41-s − 6/7·49-s − 1.64·53-s + 1.02·61-s − 1.95·67-s − 0.949·71-s + 0.468·73-s + 2.70·79-s + 7/9·81-s + 1.75·83-s − 1.27·89-s − 0.406·97-s + 1.60·99-s + 2.38·101-s + 0.788·103-s − 0.383·109-s + 0.376·113-s + 1.47·117-s − 0.181·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018091471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018091471\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 196 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 180 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.042149930378445665636952081682, −7.69535083055410428547884514559, −7.45548773579407843990416692739, −6.98892368146999663983191548049, −6.62374452663672487176916080685, −6.45389559472608344974754833294, −5.92876749744838860483562920268, −5.45122034197004194431172766501, −5.21847762066550352587387604999, −5.00265182827227807426318835578, −4.59068554177933214965126969490, −4.33664549292595917755534637961, −3.37665729420661670323515234755, −3.35893351253228519064798607102, −2.85687802117421105182169389820, −2.62651512404943490132054834736, −2.05449333845814945776335292826, −1.72320682030606371050514949664, −0.78310949072905735457864058029, −0.30615668954123824183501631935,
0.30615668954123824183501631935, 0.78310949072905735457864058029, 1.72320682030606371050514949664, 2.05449333845814945776335292826, 2.62651512404943490132054834736, 2.85687802117421105182169389820, 3.35893351253228519064798607102, 3.37665729420661670323515234755, 4.33664549292595917755534637961, 4.59068554177933214965126969490, 5.00265182827227807426318835578, 5.21847762066550352587387604999, 5.45122034197004194431172766501, 5.92876749744838860483562920268, 6.45389559472608344974754833294, 6.62374452663672487176916080685, 6.98892368146999663983191548049, 7.45548773579407843990416692739, 7.69535083055410428547884514559, 8.042149930378445665636952081682
