| L(s) = 1 | − 1.13·3-s − 2.77·5-s + 1.98·7-s − 1.70·9-s − 5.06·11-s + 4.33·13-s + 3.16·15-s − 2.00·17-s + 19-s − 2.26·21-s − 4.07·23-s + 2.72·25-s + 5.35·27-s + 0.0668·29-s + 1.97·31-s + 5.77·33-s − 5.52·35-s − 10.4·37-s − 4.94·39-s − 5.38·41-s − 9.33·43-s + 4.72·45-s + 2.29·47-s − 3.04·49-s + 2.28·51-s + 53-s + 14.0·55-s + ⋯ |
| L(s) = 1 | − 0.658·3-s − 1.24·5-s + 0.751·7-s − 0.566·9-s − 1.52·11-s + 1.20·13-s + 0.818·15-s − 0.486·17-s + 0.229·19-s − 0.494·21-s − 0.850·23-s + 0.545·25-s + 1.03·27-s + 0.0124·29-s + 0.354·31-s + 1.00·33-s − 0.933·35-s − 1.71·37-s − 0.791·39-s − 0.840·41-s − 1.42·43-s + 0.704·45-s + 0.334·47-s − 0.435·49-s + 0.320·51-s + 0.137·53-s + 1.89·55-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
| \(L(1)\) |
\(\approx\) |
\(0.6065837997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6065837997\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 - 0.0668T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 59 | \( 1 + 0.209T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 + 7.32T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
| 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
−8.274582585104682393833663874274, −7.937157645300486397245450383963, −7.01592987092415077792024024027, −6.19256950939755184380846723476, −5.29438834353710999397435309751, −4.87170805691284174964412830056, −3.85021019813305692253245145220, −3.12031151814227417587938693738, −1.89712406792363982687454415042, −0.44612207361782947663317709426,
0.44612207361782947663317709426, 1.89712406792363982687454415042, 3.12031151814227417587938693738, 3.85021019813305692253245145220, 4.87170805691284174964412830056, 5.29438834353710999397435309751, 6.19256950939755184380846723476, 7.01592987092415077792024024027, 7.937157645300486397245450383963, 8.274582585104682393833663874274
