| L(s) = 1 | − 2-s + 3-s + 4-s − 2.21·5-s − 6-s + 3.78·7-s − 8-s + 9-s + 2.21·10-s − 2.35·11-s + 12-s − 1.34·13-s − 3.78·14-s − 2.21·15-s + 16-s − 5.79·17-s − 18-s − 19-s − 2.21·20-s + 3.78·21-s + 2.35·22-s − 5.30·23-s − 24-s − 0.0720·25-s + 1.34·26-s + 27-s + 3.78·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.992·5-s − 0.408·6-s + 1.43·7-s − 0.353·8-s + 0.333·9-s + 0.701·10-s − 0.711·11-s + 0.288·12-s − 0.371·13-s − 1.01·14-s − 0.573·15-s + 0.250·16-s − 1.40·17-s − 0.235·18-s − 0.229·19-s − 0.496·20-s + 0.826·21-s + 0.502·22-s − 1.10·23-s − 0.204·24-s − 0.0144·25-s + 0.262·26-s + 0.192·27-s + 0.716·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304164380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304164380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 23 | \( 1 + 5.30T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 + 0.138T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 7.71T + 61T^{2} \) |
| 67 | \( 1 - 7.18T + 67T^{2} \) |
| 71 | \( 1 - 3.36T + 71T^{2} \) |
| 73 | \( 1 - 7.36T + 73T^{2} \) |
| 79 | \( 1 - 9.58T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−8.037650229279163270617013519445, −7.77682539538614978760127894762, −6.97611151810270695605951453304, −6.12159987749899053460899456500, −4.95513798238667047690654387152, −4.45281441907261851646468445403, −3.65657321878457384210657440444, −2.43851631665970976588695036919, −1.99419960986636153246292206290, −0.63041395774518832466210163997,
0.63041395774518832466210163997, 1.99419960986636153246292206290, 2.43851631665970976588695036919, 3.65657321878457384210657440444, 4.45281441907261851646468445403, 4.95513798238667047690654387152, 6.12159987749899053460899456500, 6.97611151810270695605951453304, 7.77682539538614978760127894762, 8.037650229279163270617013519445
