| L(s) = 1 | + 2.78·2-s − 3-s + 5.78·4-s − 0.430·5-s − 2.78·6-s + 7-s + 10.5·8-s + 9-s − 1.20·10-s − 4.67·11-s − 5.78·12-s − 4.61·13-s + 2.78·14-s + 0.430·15-s + 17.8·16-s + 5.52·17-s + 2.78·18-s − 19-s − 2.49·20-s − 21-s − 13.0·22-s − 1.98·23-s − 10.5·24-s − 4.81·25-s − 12.8·26-s − 27-s + 5.78·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.89·4-s − 0.192·5-s − 1.13·6-s + 0.377·7-s + 3.72·8-s + 0.333·9-s − 0.380·10-s − 1.40·11-s − 1.66·12-s − 1.27·13-s + 0.745·14-s + 0.111·15-s + 4.46·16-s + 1.33·17-s + 0.657·18-s − 0.229·19-s − 0.557·20-s − 0.218·21-s − 2.77·22-s − 0.413·23-s − 2.15·24-s − 0.962·25-s − 2.52·26-s − 0.192·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.540379174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.540379174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 5 | \( 1 + 0.430T + 5T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 + 0.487T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 - 0.358T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 2.40T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.73545234561521323868513393262, −10.63897997137810765600274224106, −10.12936485431719298845948109650, −7.79178210570273609880653358134, −7.38524109678066328852007582063, −6.06254589167289660002334506624, −5.26071689981886757647543476142, −4.65810458986392745636658233734, −3.34869610518073482208802500588, −2.12990077093083298421564681007,
2.12990077093083298421564681007, 3.34869610518073482208802500588, 4.65810458986392745636658233734, 5.26071689981886757647543476142, 6.06254589167289660002334506624, 7.38524109678066328852007582063, 7.79178210570273609880653358134, 10.12936485431719298845948109650, 10.63897997137810765600274224106, 11.73545234561521323868513393262
