| L(s) = 1 | − 1.42·2-s + 3-s + 0.0280·4-s − 0.882·5-s − 1.42·6-s + 4.13·7-s + 2.80·8-s + 9-s + 1.25·10-s − 6.10·11-s + 0.0280·12-s − 13-s − 5.88·14-s − 0.882·15-s − 4.05·16-s − 6.47·17-s − 1.42·18-s − 7.20·19-s − 0.0247·20-s + 4.13·21-s + 8.69·22-s + 6.81·23-s + 2.80·24-s − 4.22·25-s + 1.42·26-s + 27-s + 0.116·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 0.0140·4-s − 0.394·5-s − 0.581·6-s + 1.56·7-s + 0.992·8-s + 0.333·9-s + 0.397·10-s − 1.84·11-s + 0.00810·12-s − 0.277·13-s − 1.57·14-s − 0.227·15-s − 1.01·16-s − 1.56·17-s − 0.335·18-s − 1.65·19-s − 0.00553·20-s + 0.901·21-s + 1.85·22-s + 1.42·23-s + 0.573·24-s − 0.844·25-s + 0.279·26-s + 0.192·27-s + 0.0219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9635243530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9635243530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 + 0.882T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 + 4.58T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 9.78T + 79T^{2} \) |
| 83 | \( 1 - 6.60T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.441545852653914483450041113558, −7.958461689457053126485141343441, −7.45559942578699303361008527778, −6.56923252872189543733973468871, −5.08662085204693224292976804892, −4.75982624916312379753106045754, −4.01308242307779209710789201445, −2.42867192037068900867353304938, −2.06767928441388269026992383977, −0.62309630277425318932028891222,
0.62309630277425318932028891222, 2.06767928441388269026992383977, 2.42867192037068900867353304938, 4.01308242307779209710789201445, 4.75982624916312379753106045754, 5.08662085204693224292976804892, 6.56923252872189543733973468871, 7.45559942578699303361008527778, 7.958461689457053126485141343441, 8.441545852653914483450041113558
