L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s + 2·11-s + 2·12-s − 13-s − 4·16-s − 2·17-s + 2·18-s + 5·19-s + 4·22-s + 6·23-s − 2·26-s + 27-s + 10·29-s + 3·31-s − 8·32-s + 2·33-s − 4·34-s + 2·36-s + 2·37-s + 10·38-s − 39-s + 8·41-s + 43-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s − 16-s − 0.485·17-s + 0.471·18-s + 1.14·19-s + 0.852·22-s + 1.25·23-s − 0.392·26-s + 0.192·27-s + 1.85·29-s + 0.538·31-s − 1.41·32-s + 0.348·33-s − 0.685·34-s + 1/3·36-s + 0.328·37-s + 1.62·38-s − 0.160·39-s + 1.24·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.294752453\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.294752453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.569123090874178563709672380540, −7.60923711989545913207621904275, −6.82044341445639010249085887854, −6.28354708317492806397521188452, −5.28452479922674377650498409826, −4.65405693802966717273237919028, −3.98196758650784752573254151818, −3.04023520942509040038527723076, −2.56863529572190940234034504682, −1.13430400006930951329029124705,
1.13430400006930951329029124705, 2.56863529572190940234034504682, 3.04023520942509040038527723076, 3.98196758650784752573254151818, 4.65405693802966717273237919028, 5.28452479922674377650498409826, 6.28354708317492806397521188452, 6.82044341445639010249085887854, 7.60923711989545913207621904275, 8.569123090874178563709672380540