| L(s) = 1 | + 1.75·2-s + 0.822·3-s + 1.07·4-s + 1.44·6-s − 3.28·7-s − 1.61·8-s − 2.32·9-s + 1.18·11-s + 0.886·12-s − 3.78·13-s − 5.76·14-s − 4.99·16-s + 2.75·17-s − 4.07·18-s − 2.70·21-s + 2.07·22-s − 3.31·23-s − 1.33·24-s − 6.63·26-s − 4.37·27-s − 3.54·28-s + 9.75·29-s − 1.66·31-s − 5.52·32-s + 0.970·33-s + 4.83·34-s − 2.50·36-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.474·3-s + 0.538·4-s + 0.589·6-s − 1.24·7-s − 0.572·8-s − 0.774·9-s + 0.355·11-s + 0.255·12-s − 1.04·13-s − 1.54·14-s − 1.24·16-s + 0.668·17-s − 0.960·18-s − 0.589·21-s + 0.441·22-s − 0.691·23-s − 0.271·24-s − 1.30·26-s − 0.842·27-s − 0.669·28-s + 1.81·29-s − 0.298·31-s − 0.976·32-s + 0.169·33-s + 0.828·34-s − 0.417·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.645990510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645990510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 - 0.822T + 3T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 0.0828T + 53T^{2} \) |
| 59 | \( 1 - 4.80T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 - 8.59T + 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.69545536198888664585931911820, −6.66953374707668847458071596310, −6.35948128407736924328100768190, −5.56586179052967203003588845859, −5.00162046199522307888637610092, −4.04807371023393623198348849440, −3.55270328367723371021289702009, −2.74852342826752691252856998574, −2.40208267055506467878396092865, −0.59469184881650346758245409117,
0.59469184881650346758245409117, 2.40208267055506467878396092865, 2.74852342826752691252856998574, 3.55270328367723371021289702009, 4.04807371023393623198348849440, 5.00162046199522307888637610092, 5.56586179052967203003588845859, 6.35948128407736924328100768190, 6.66953374707668847458071596310, 7.69545536198888664585931911820
