| L(s) = 1 | + 2.24·3-s − 5-s + 3.06·7-s + 2.03·9-s + 1.64·11-s + 4.87·13-s − 2.24·15-s + 3.34·17-s + 2.83·19-s + 6.88·21-s + 8.66·23-s + 25-s − 2.16·27-s − 3.80·29-s − 2.88·31-s + 3.68·33-s − 3.06·35-s − 6.42·37-s + 10.9·39-s + 3.28·41-s − 3.64·43-s − 2.03·45-s + 0.752·47-s + 2.40·49-s + 7.49·51-s + 1.41·53-s − 1.64·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 0.447·5-s + 1.15·7-s + 0.678·9-s + 0.494·11-s + 1.35·13-s − 0.579·15-s + 0.810·17-s + 0.650·19-s + 1.50·21-s + 1.80·23-s + 0.200·25-s − 0.416·27-s − 0.706·29-s − 0.518·31-s + 0.640·33-s − 0.518·35-s − 1.05·37-s + 1.75·39-s + 0.512·41-s − 0.555·43-s − 0.303·45-s + 0.109·47-s + 0.343·49-s + 1.04·51-s + 0.194·53-s − 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.138276070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.138276070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 2.83T + 19T^{2} \) |
| 23 | \( 1 - 8.66T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 2.88T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 0.752T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 + 0.454T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 4.55T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−8.171781772409404090804708879250, −7.49039961329684746844220995324, −7.03350974871399955909619579059, −5.83814970798589649761279849118, −5.14697430616112420548959602583, −4.22132343723885110160505630330, −3.49484672871960631910384951942, −3.00864278728424781130072842564, −1.76578745653720922152945951517, −1.13887767084633035989377945430,
1.13887767084633035989377945430, 1.76578745653720922152945951517, 3.00864278728424781130072842564, 3.49484672871960631910384951942, 4.22132343723885110160505630330, 5.14697430616112420548959602583, 5.83814970798589649761279849118, 7.03350974871399955909619579059, 7.49039961329684746844220995324, 8.171781772409404090804708879250
