| L(s) = 1 | − 2.54·3-s − 2.87·7-s + 3.46·9-s + 4.99·11-s − 0.149·13-s + 6.23·17-s − 1.90·19-s + 7.32·21-s − 4.35·23-s − 1.19·27-s − 8.93·29-s − 8.99·31-s − 12.7·33-s + 2.56·37-s + 0.379·39-s + 7.99·41-s − 4.54·43-s − 9.68·47-s + 1.28·49-s − 15.8·51-s − 1.52·53-s + 4.85·57-s + 11.4·59-s − 3.55·61-s − 9.98·63-s − 4.08·67-s + 11.0·69-s + ⋯ |
| L(s) = 1 | − 1.46·3-s − 1.08·7-s + 1.15·9-s + 1.50·11-s − 0.0414·13-s + 1.51·17-s − 0.437·19-s + 1.59·21-s − 0.908·23-s − 0.229·27-s − 1.65·29-s − 1.61·31-s − 2.21·33-s + 0.421·37-s + 0.0608·39-s + 1.24·41-s − 0.692·43-s − 1.41·47-s + 0.184·49-s − 2.21·51-s − 0.209·53-s + 0.642·57-s + 1.49·59-s − 0.455·61-s − 1.25·63-s − 0.498·67-s + 1.33·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 4.99T + 11T^{2} \) |
| 13 | \( 1 + 0.149T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 + 4.54T + 43T^{2} \) |
| 47 | \( 1 + 9.68T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
| 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
−9.725276904353479422998217414847, −9.006012014627825514812476116523, −7.60548128652200670080916526951, −6.77937199092904670089791612306, −6.01291076837440709499011047570, −5.56643650734157994508055313524, −4.21577703712064417281421256843, −3.40605365645506965289013402652, −1.49123960040188935965338824048, 0,
1.49123960040188935965338824048, 3.40605365645506965289013402652, 4.21577703712064417281421256843, 5.56643650734157994508055313524, 6.01291076837440709499011047570, 6.77937199092904670089791612306, 7.60548128652200670080916526951, 9.006012014627825514812476116523, 9.725276904353479422998217414847
