| L(s) = 1 | − 8·3-s + 4·7-s + 37·9-s − 12·11-s − 58·13-s − 66·17-s + 100·19-s − 32·21-s − 132·23-s − 80·27-s + 90·29-s + 152·31-s + 96·33-s − 34·37-s + 464·39-s − 438·41-s + 32·43-s + 204·47-s − 327·49-s + 528·51-s + 222·53-s − 800·57-s − 420·59-s − 902·61-s + 148·63-s − 1.02e3·67-s + 1.05e3·69-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.215·7-s + 1.37·9-s − 0.328·11-s − 1.23·13-s − 0.941·17-s + 1.20·19-s − 0.332·21-s − 1.19·23-s − 0.570·27-s + 0.576·29-s + 0.880·31-s + 0.506·33-s − 0.151·37-s + 1.90·39-s − 1.66·41-s + 0.113·43-s + 0.633·47-s − 0.953·49-s + 1.44·51-s + 0.575·53-s − 1.85·57-s − 0.926·59-s − 1.89·61-s + 0.295·63-s − 1.86·67-s + 1.84·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5581072554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5581072554\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 902 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 72 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1106 T + p^{3} T^{2} \) |
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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
−9.219970559659714382697263648511, −8.070614692104420264303434326594, −7.28527225412483697510826710117, −6.51780775109043327009633611494, −5.74993507081363744649661733850, −4.92660297792051926480789744183, −4.44964776425106446974397458796, −2.94709819939421504990515018060, −1.68252109530772182000652341089, −0.38807519433889525814483574980,
0.38807519433889525814483574980, 1.68252109530772182000652341089, 2.94709819939421504990515018060, 4.44964776425106446974397458796, 4.92660297792051926480789744183, 5.74993507081363744649661733850, 6.51780775109043327009633611494, 7.28527225412483697510826710117, 8.070614692104420264303434326594, 9.219970559659714382697263648511
