| L(s) = 1 | − 617·3-s + 4.09e3·4-s − 2.31e5·7-s − 1.50e5·9-s − 3.19e6·11-s − 2.52e6·12-s + 1.67e7·16-s − 3.06e7·17-s + 1.42e8·21-s − 2.31e8·23-s + 2.44e8·25-s + 4.20e8·27-s − 9.48e8·28-s + 9.91e8·29-s − 1.01e8·31-s + 1.97e9·33-s − 6.17e8·36-s − 5.01e9·37-s − 9.47e9·41-s − 1.31e10·44-s − 1.03e10·48-s + 3.97e10·49-s + 1.88e10·51-s − 4.86e10·59-s + 8.64e10·61-s + 3.49e10·63-s + 6.87e10·64-s + ⋯ |
| L(s) = 1 | − 0.846·3-s + 4-s − 1.96·7-s − 0.283·9-s − 1.80·11-s − 0.846·12-s + 16-s − 1.26·17-s + 1.66·21-s − 1.56·23-s + 25-s + 1.08·27-s − 1.96·28-s + 1.66·29-s − 0.114·31-s + 1.52·33-s − 0.283·36-s − 1.95·37-s − 1.99·41-s − 1.80·44-s − 0.846·48-s + 2.87·49-s + 1.07·51-s − 1.15·59-s + 1.67·61-s + 0.558·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.4046423177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4046423177\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - p^{6} T \) |
| good | 2 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 3 | \( 1 + 617 T + p^{12} T^{2} \) |
| 5 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( 1 + 231577 T + p^{12} T^{2} \) |
| 11 | \( 1 + 3198553 T + p^{12} T^{2} \) |
| 13 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 17 | \( 1 + 30626737 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( 1 + 231011422 T + p^{12} T^{2} \) |
| 29 | \( 1 - 991523567 T + p^{12} T^{2} \) |
| 31 | \( 1 + 101624713 T + p^{12} T^{2} \) |
| 37 | \( 1 + 5012228257 T + p^{12} T^{2} \) |
| 41 | \( 1 + 9474786718 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 48671434393 T + p^{12} T^{2} \) |
| 61 | \( 1 - 86434023647 T + p^{12} T^{2} \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 73 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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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
−11.94992865096724962153336969633, −10.55618045907570437946752530148, −10.22750115083136142711227326600, −8.464549739067672590445757732086, −6.88177943401389532750221930900, −6.30861469594715554447479931013, −5.22344863319334557795943337861, −3.23404257909401156381187209558, −2.38483259894561654843019876340, −0.30862920651179148533908835116,
0.30862920651179148533908835116, 2.38483259894561654843019876340, 3.23404257909401156381187209558, 5.22344863319334557795943337861, 6.30861469594715554447479931013, 6.88177943401389532750221930900, 8.464549739067672590445757732086, 10.22750115083136142711227326600, 10.55618045907570437946752530148, 11.94992865096724962153336969633
