| L(s) = 1 | − 2-s + 7·3-s − 7·4-s − 7·6-s + 15·8-s + 22·9-s − 43·11-s − 49·12-s − 28·13-s + 41·16-s + 91·17-s − 22·18-s + 35·19-s + 43·22-s − 162·23-s + 105·24-s + 28·26-s − 35·27-s + 160·29-s − 42·31-s − 161·32-s − 301·33-s − 91·34-s − 154·36-s + 314·37-s − 35·38-s − 196·39-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.34·3-s − 7/8·4-s − 0.476·6-s + 0.662·8-s + 0.814·9-s − 1.17·11-s − 1.17·12-s − 0.597·13-s + 0.640·16-s + 1.29·17-s − 0.288·18-s + 0.422·19-s + 0.416·22-s − 1.46·23-s + 0.893·24-s + 0.211·26-s − 0.249·27-s + 1.02·29-s − 0.243·31-s − 0.889·32-s − 1.58·33-s − 0.459·34-s − 0.712·36-s + 1.39·37-s − 0.149·38-s − 0.804·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.973053811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973053811\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 91 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 160 T + p^{3} T^{2} \) |
| 31 | \( 1 + 42 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 203 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 196 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 280 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 141 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 - 777 T + p^{3} T^{2} \) |
| 89 | \( 1 - 945 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1246 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.357772744386978506043115370372, −8.440385762349886352076975666596, −7.85109417693789969047681975892, −7.51785632025509189070530287368, −5.87173198379575783255338619726, −4.98043022329784581496043086190, −3.98863782569926096523343680733, −3.07325468750169686524040581622, −2.15660129672442394905879410010, −0.70554732074754400172143118691,
0.70554732074754400172143118691, 2.15660129672442394905879410010, 3.07325468750169686524040581622, 3.98863782569926096523343680733, 4.98043022329784581496043086190, 5.87173198379575783255338619726, 7.51785632025509189070530287368, 7.85109417693789969047681975892, 8.440385762349886352076975666596, 9.357772744386978506043115370372
