| L(s) = 1 | − 2-s − 1.87·3-s + 1.87·6-s + 8-s + 2.53·9-s − 16-s − 1.87·17-s − 2.53·18-s + 0.347·23-s − 1.87·24-s + 25-s − 2.87·27-s − 1.87·31-s + 1.87·34-s + 0.347·37-s + 1.53·41-s − 0.347·46-s + 1.87·48-s + 49-s − 50-s + 3.53·51-s − 1.87·53-s + 2.87·54-s + 0.347·59-s + 1.87·62-s + 64-s + 1.53·67-s + ⋯ |
| L(s) = 1 | − 2-s − 1.87·3-s + 1.87·6-s + 8-s + 2.53·9-s − 16-s − 1.87·17-s − 2.53·18-s + 0.347·23-s − 1.87·24-s + 25-s − 2.87·27-s − 1.87·31-s + 1.87·34-s + 0.347·37-s + 1.53·41-s − 0.347·46-s + 1.87·48-s + 49-s − 50-s + 3.53·51-s − 1.87·53-s + 2.87·54-s + 0.347·59-s + 1.87·62-s + 64-s + 1.53·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1607 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1607 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2643138118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2643138118\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1607 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 + 1.87T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.347T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.87T + T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 - 1.53T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 - 0.347T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.53T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.501406964450930898681668678477, −9.127286992037890997595538173084, −7.964826619218766184799269713164, −7.02975673050162125746815794263, −6.58586074109333343961397607992, −5.51122618681680652642453779464, −4.76660485996989065534298728072, −4.06553163790950539769540800055, −1.96913379618923402435321359855, −0.68365409226508332719049000909,
0.68365409226508332719049000909, 1.96913379618923402435321359855, 4.06553163790950539769540800055, 4.76660485996989065534298728072, 5.51122618681680652642453779464, 6.58586074109333343961397607992, 7.02975673050162125746815794263, 7.964826619218766184799269713164, 9.127286992037890997595538173084, 9.501406964450930898681668678477
