| L(s) = 1 | − 3-s − 1.39·5-s − 3.28·7-s + 9-s − 11-s − 13-s + 1.39·15-s − 2.67·17-s + 3.28·19-s + 3.28·21-s − 3.28·23-s − 3.06·25-s − 27-s + 1.89·29-s + 3.39·31-s + 33-s + 4.56·35-s + 4·37-s + 39-s + 5.28·41-s − 1.89·43-s − 1.39·45-s − 7.34·47-s + 3.78·49-s + 2.67·51-s + 11.3·53-s + 1.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.622·5-s − 1.24·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.359·15-s − 0.648·17-s + 0.753·19-s + 0.716·21-s − 0.684·23-s − 0.613·25-s − 0.192·27-s + 0.351·29-s + 0.609·31-s + 0.174·33-s + 0.771·35-s + 0.657·37-s + 0.160·39-s + 0.825·41-s − 0.288·43-s − 0.207·45-s − 1.07·47-s + 0.540·49-s + 0.374·51-s + 1.55·53-s + 0.187·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7301556097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7301556097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + 1.89T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.21T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 - 3.32T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.621193585757189847023852368889, −8.474210794604485201723388886115, −7.66585036465243106031576486946, −6.84918531083428140145052038223, −6.19438549589984351985734384391, −5.30863172848548754088312400544, −4.28508939984462036671859222591, −3.47626008132332711662296162767, −2.38419618032047628741324207357, −0.58025398588957622052670049695,
0.58025398588957622052670049695, 2.38419618032047628741324207357, 3.47626008132332711662296162767, 4.28508939984462036671859222591, 5.30863172848548754088312400544, 6.19438549589984351985734384391, 6.84918531083428140145052038223, 7.66585036465243106031576486946, 8.474210794604485201723388886115, 9.621193585757189847023852368889
