L(s) = 1 | − 3-s − 3·7-s + 9-s − 2·11-s − 3·13-s + 6·17-s + 7·19-s + 3·21-s − 6·23-s − 27-s − 2·29-s + 5·31-s + 2·33-s + 10·37-s + 3·39-s + 12·41-s − 3·43-s + 10·47-s + 2·49-s − 6·51-s − 7·57-s + 6·59-s − 13·61-s − 3·63-s − 7·67-s + 6·69-s + 4·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.832·13-s + 1.45·17-s + 1.60·19-s + 0.654·21-s − 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.898·31-s + 0.348·33-s + 1.64·37-s + 0.480·39-s + 1.87·41-s − 0.457·43-s + 1.45·47-s + 2/7·49-s − 0.840·51-s − 0.927·57-s + 0.781·59-s − 1.66·61-s − 0.377·63-s − 0.855·67-s + 0.722·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.046299298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.046299298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.882613299713563149760910871277, −9.250912384445108460730408413088, −7.66165741262556570263007689206, −7.55015404230883423512821682767, −6.16960063995750489450981625407, −5.71322079987385705135293329202, −4.69169079872189797958426439923, −3.50574880723700274234440164435, −2.58093474510021419032571611233, −0.77604421210661452476716259489,
0.77604421210661452476716259489, 2.58093474510021419032571611233, 3.50574880723700274234440164435, 4.69169079872189797958426439923, 5.71322079987385705135293329202, 6.16960063995750489450981625407, 7.55015404230883423512821682767, 7.66165741262556570263007689206, 9.250912384445108460730408413088, 9.882613299713563149760910871277