| L(s) = 1 | + 0.476·3-s − 2.17·5-s − 2.77·9-s − 0.849·11-s + 0.662·13-s − 1.03·15-s − 5.15·17-s − 6.79·19-s − 0.144·23-s − 0.259·25-s − 2.74·27-s − 9.46·29-s + 8.50·31-s − 0.404·33-s − 7.75·37-s + 0.315·39-s − 41-s − 2.77·43-s + 6.03·45-s + 3.45·47-s − 2.45·51-s + 7.28·53-s + 1.85·55-s − 3.23·57-s + 9.84·59-s + 9.68·61-s − 1.44·65-s + ⋯ |
| L(s) = 1 | + 0.274·3-s − 0.973·5-s − 0.924·9-s − 0.256·11-s + 0.183·13-s − 0.267·15-s − 1.25·17-s − 1.55·19-s − 0.0301·23-s − 0.0518·25-s − 0.528·27-s − 1.75·29-s + 1.52·31-s − 0.0704·33-s − 1.27·37-s + 0.0504·39-s − 0.156·41-s − 0.422·43-s + 0.900·45-s + 0.504·47-s − 0.343·51-s + 1.00·53-s + 0.249·55-s − 0.428·57-s + 1.28·59-s + 1.23·61-s − 0.178·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6584989810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6584989810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.476T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 11 | \( 1 + 0.849T + 11T^{2} \) |
| 13 | \( 1 - 0.662T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 + 6.79T + 19T^{2} \) |
| 23 | \( 1 + 0.144T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 8.50T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 - 9.84T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 5.94T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−8.011324282432508514460512031144, −7.14424414480618562801396985493, −6.55901834853862673921369650120, −5.76288069704130459598975487319, −4.97955406241519360875964497577, −4.06363016459030620119526648571, −3.70075633018236602158977456899, −2.61357634979203658088589946779, −1.99740179631202251100924518413, −0.36952656458875077245201905468,
0.36952656458875077245201905468, 1.99740179631202251100924518413, 2.61357634979203658088589946779, 3.70075633018236602158977456899, 4.06363016459030620119526648571, 4.97955406241519360875964497577, 5.76288069704130459598975487319, 6.55901834853862673921369650120, 7.14424414480618562801396985493, 8.011324282432508514460512031144
