L(s) = 1 | − 3-s − 2·9-s − 2·11-s + 4·13-s − 6·19-s − 3·23-s + 5·27-s − 3·29-s + 2·33-s + 12·37-s − 4·39-s + 7·41-s + 9·43-s + 6·53-s + 6·57-s + 10·59-s − 5·61-s − 11·67-s + 3·69-s − 10·71-s − 8·73-s + 6·79-s + 81-s − 3·83-s + 3·87-s − 17·89-s − 2·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.603·11-s + 1.10·13-s − 1.37·19-s − 0.625·23-s + 0.962·27-s − 0.557·29-s + 0.348·33-s + 1.97·37-s − 0.640·39-s + 1.09·41-s + 1.37·43-s + 0.824·53-s + 0.794·57-s + 1.30·59-s − 0.640·61-s − 1.34·67-s + 0.361·69-s − 1.18·71-s − 0.936·73-s + 0.675·79-s + 1/9·81-s − 0.329·83-s + 0.321·87-s − 1.80·89-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.38512903112562573781308222504, −6.38287703738034442874881454531, −5.94659198602379135893934911843, −5.54699687778848126341040773363, −4.43644513492105106421304361074, −4.02732232935475872364087404274, −2.90335523080399085362963137251, −2.26095826726702758122777200067, −1.05023506035652117450715144214, 0,
1.05023506035652117450715144214, 2.26095826726702758122777200067, 2.90335523080399085362963137251, 4.02732232935475872364087404274, 4.43644513492105106421304361074, 5.54699687778848126341040773363, 5.94659198602379135893934911843, 6.38287703738034442874881454531, 7.38512903112562573781308222504