| L(s) = 1 | + 0.407·3-s + 2.35·5-s − 3.33·7-s − 2.83·9-s + 4.73·11-s + 4.54·13-s + 0.957·15-s − 8.21·17-s − 0.775·19-s − 1.35·21-s + 0.527·25-s − 2.37·27-s − 3.80·29-s − 3.89·31-s + 1.92·33-s − 7.82·35-s − 3.79·37-s + 1.85·39-s − 6.95·41-s − 11.9·43-s − 6.66·45-s − 0.399·47-s + 4.09·49-s − 3.34·51-s + 5.42·53-s + 11.1·55-s − 0.316·57-s + ⋯ |
| L(s) = 1 | + 0.235·3-s + 1.05·5-s − 1.25·7-s − 0.944·9-s + 1.42·11-s + 1.26·13-s + 0.247·15-s − 1.99·17-s − 0.177·19-s − 0.296·21-s + 0.105·25-s − 0.457·27-s − 0.705·29-s − 0.699·31-s + 0.335·33-s − 1.32·35-s − 0.623·37-s + 0.296·39-s − 1.08·41-s − 1.82·43-s − 0.993·45-s − 0.0582·47-s + 0.584·49-s − 0.468·51-s + 0.744·53-s + 1.49·55-s − 0.0418·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.407T + 3T^{2} \) |
| 5 | \( 1 - 2.35T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 8.21T + 17T^{2} \) |
| 19 | \( 1 + 0.775T + 19T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 0.399T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 0.371T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 2.59T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 3.62T + 83T^{2} \) |
| 89 | \( 1 + 1.67T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.477976999608612072695330715677, −6.82725245765837875350107590775, −6.61754621650527089444014999294, −6.00521395157124684236608654605, −5.25605629811764903982431750306, −3.91559301121061497936365032366, −3.49715712901354303674407711849, −2.38708301404526945322259720543, −1.59856207619618760058184258617, 0,
1.59856207619618760058184258617, 2.38708301404526945322259720543, 3.49715712901354303674407711849, 3.91559301121061497936365032366, 5.25605629811764903982431750306, 6.00521395157124684236608654605, 6.61754621650527089444014999294, 6.82725245765837875350107590775, 8.477976999608612072695330715677
