| L(s) = 1 | − 1.58·3-s + 2.56·5-s + 3.40·7-s − 0.489·9-s + 2.40·11-s + 2.00·13-s − 4.06·15-s + 17-s + 3.89·19-s − 5.39·21-s − 3.11·23-s + 1.59·25-s + 5.52·27-s + 5.87·29-s + 3.36·31-s − 3.80·33-s + 8.74·35-s + 4.17·37-s − 3.18·39-s − 2.04·41-s − 4.64·43-s − 1.25·45-s − 4.53·47-s + 4.59·49-s − 1.58·51-s + 4.71·53-s + 6.16·55-s + ⋯ |
| L(s) = 1 | − 0.914·3-s + 1.14·5-s + 1.28·7-s − 0.163·9-s + 0.724·11-s + 0.557·13-s − 1.05·15-s + 0.242·17-s + 0.892·19-s − 1.17·21-s − 0.649·23-s + 0.319·25-s + 1.06·27-s + 1.09·29-s + 0.603·31-s − 0.662·33-s + 1.47·35-s + 0.686·37-s − 0.509·39-s − 0.319·41-s − 0.707·43-s − 0.187·45-s − 0.660·47-s + 0.655·49-s − 0.221·51-s + 0.647·53-s + 0.831·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.271202849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.271202849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 - 9.35T + 67T^{2} \) |
| 71 | \( 1 + 0.402T + 71T^{2} \) |
| 73 | \( 1 + 0.0127T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.395803664774267351209104411670, −7.82985425601640185082040463095, −6.61734054385995583911024660586, −6.24261444047360386317437278714, −5.39940211640140651872183928799, −5.01366302855059100939647299593, −4.04363538928652605034403205857, −2.81087536125840125966027818158, −1.70783902974197103369134099862, −1.00036694224404842266287327170,
1.00036694224404842266287327170, 1.70783902974197103369134099862, 2.81087536125840125966027818158, 4.04363538928652605034403205857, 5.01366302855059100939647299593, 5.39940211640140651872183928799, 6.24261444047360386317437278714, 6.61734054385995583911024660586, 7.82985425601640185082040463095, 8.395803664774267351209104411670
