| L(s) = 1 | − 1.30·3-s − 2.30·5-s + 2.60·7-s − 1.30·9-s − 4.30·11-s + 1.30·13-s + 3·15-s + 7.21·17-s + 6·19-s − 3.39·21-s − 4.69·23-s + 0.302·25-s + 5.60·27-s − 6.69·29-s + 1.69·31-s + 5.60·33-s − 6·35-s + 37-s − 1.69·39-s − 3.30·41-s + 8.60·43-s + 3.00·45-s + 3.39·47-s − 0.211·49-s − 9.39·51-s − 7.21·53-s + 9.90·55-s + ⋯ |
| L(s) = 1 | − 0.752·3-s − 1.02·5-s + 0.984·7-s − 0.434·9-s − 1.29·11-s + 0.361·13-s + 0.774·15-s + 1.74·17-s + 1.37·19-s − 0.740·21-s − 0.979·23-s + 0.0605·25-s + 1.07·27-s − 1.24·29-s + 0.304·31-s + 0.975·33-s − 1.01·35-s + 0.164·37-s − 0.271·39-s − 0.515·41-s + 1.31·43-s + 0.447·45-s + 0.495·47-s − 0.0301·49-s − 1.31·51-s − 0.990·53-s + 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 4.30T + 73T^{2} \) |
| 79 | \( 1 + 4.69T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 0.788T + 97T^{2} \) |
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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
−8.236892518577937037428452871110, −7.78490977879740658181867648544, −7.42362953664918695756571454908, −5.91390320727409415639399791165, −5.49240628919288568642473632033, −4.76425622305794922930154831191, −3.71376498573645612762292154340, −2.83320563003836383221363635042, −1.30150012601447988763520962306, 0,
1.30150012601447988763520962306, 2.83320563003836383221363635042, 3.71376498573645612762292154340, 4.76425622305794922930154831191, 5.49240628919288568642473632033, 5.91390320727409415639399791165, 7.42362953664918695756571454908, 7.78490977879740658181867648544, 8.236892518577937037428452871110
