| L(s) = 1 | − 2.40·3-s − 3.21·5-s + 0.104·7-s + 2.79·9-s + 2.96·11-s + 0.989·13-s + 7.73·15-s − 6.40·17-s + 0.260·19-s − 0.250·21-s − 2.04·23-s + 5.31·25-s + 0.488·27-s − 1.24·29-s + 2.01·31-s − 7.14·33-s − 0.334·35-s − 6.64·37-s − 2.38·39-s − 6.12·41-s − 0.561·43-s − 8.98·45-s + 7.02·47-s − 6.98·49-s + 15.4·51-s + 6.32·53-s − 9.53·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 1.43·5-s + 0.0393·7-s + 0.932·9-s + 0.895·11-s + 0.274·13-s + 1.99·15-s − 1.55·17-s + 0.0598·19-s − 0.0546·21-s − 0.426·23-s + 1.06·25-s + 0.0940·27-s − 0.231·29-s + 0.362·31-s − 1.24·33-s − 0.0564·35-s − 1.09·37-s − 0.381·39-s − 0.956·41-s − 0.0856·43-s − 1.33·45-s + 1.02·47-s − 0.998·49-s + 2.16·51-s + 0.869·53-s − 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 - 0.104T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 0.989T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 - 0.260T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 + 0.561T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 + 1.87T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 0.733T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 3.74T + 97T^{2} \) |
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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
−7.20889448828184294315952610831, −6.77978700031987495368627822227, −6.23145760522181664776410724311, −5.33881073617967910797079882781, −4.63663937728801660359454476445, −4.05245668127545161329406190925, −3.43087226585171820029776729325, −2.06308514052876348079016784353, −0.847977281689962081749955485346, 0,
0.847977281689962081749955485346, 2.06308514052876348079016784353, 3.43087226585171820029776729325, 4.05245668127545161329406190925, 4.63663937728801660359454476445, 5.33881073617967910797079882781, 6.23145760522181664776410724311, 6.77978700031987495368627822227, 7.20889448828184294315952610831
