| L(s) = 1 | + 0.667·3-s − 3.08·5-s − 2.50·7-s − 2.55·9-s − 1.87·11-s − 0.792·13-s − 2.06·15-s − 3.96·17-s − 19-s − 1.67·21-s + 0.00275·23-s + 4.53·25-s − 3.70·27-s + 4.17·29-s − 0.345·31-s − 1.25·33-s + 7.74·35-s − 5.04·37-s − 0.529·39-s − 3.80·41-s + 0.706·43-s + 7.88·45-s − 7.21·47-s − 0.704·49-s − 2.64·51-s − 53-s + 5.80·55-s + ⋯ |
| L(s) = 1 | + 0.385·3-s − 1.38·5-s − 0.948·7-s − 0.851·9-s − 0.566·11-s − 0.219·13-s − 0.532·15-s − 0.960·17-s − 0.229·19-s − 0.365·21-s + 0.000575·23-s + 0.907·25-s − 0.713·27-s + 0.774·29-s − 0.0620·31-s − 0.218·33-s + 1.30·35-s − 0.830·37-s − 0.0847·39-s − 0.593·41-s + 0.107·43-s + 1.17·45-s − 1.05·47-s − 0.100·49-s − 0.370·51-s − 0.137·53-s + 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4951876165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4951876165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 0.667T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 0.792T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 23 | \( 1 - 0.00275T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 0.345T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 0.706T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 - 6.54T + 83T^{2} \) |
| 89 | \( 1 - 1.84T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
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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
−8.351795800139657579460935453605, −7.900979300288572862382245483636, −6.96751910160563039475968269104, −6.46057124068905027802208134323, −5.39137550727204035152499371691, −4.56256266474794761383821061856, −3.65408723654866383692260760362, −3.11426822284469462941605232464, −2.24009171562227530093229104533, −0.36578558831701432270313112551,
0.36578558831701432270313112551, 2.24009171562227530093229104533, 3.11426822284469462941605232464, 3.65408723654866383692260760362, 4.56256266474794761383821061856, 5.39137550727204035152499371691, 6.46057124068905027802208134323, 6.96751910160563039475968269104, 7.900979300288572862382245483636, 8.351795800139657579460935453605
