| L(s) = 1 | + 3.26·3-s − 5-s + 2.98·7-s + 7.66·9-s − 5.67·11-s + 0.534·13-s − 3.26·15-s − 0.983·17-s + 3.80·19-s + 9.74·21-s + 6.19·23-s + 25-s + 15.2·27-s + 29-s + 6.61·31-s − 18.5·33-s − 2.98·35-s − 1.66·37-s + 1.74·39-s − 6.40·41-s − 9.02·43-s − 7.66·45-s + 4.74·47-s + 1.90·49-s − 3.21·51-s − 11.1·53-s + 5.67·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s − 0.447·5-s + 1.12·7-s + 2.55·9-s − 1.71·11-s + 0.148·13-s − 0.843·15-s − 0.238·17-s + 0.871·19-s + 2.12·21-s + 1.29·23-s + 0.200·25-s + 2.92·27-s + 0.185·29-s + 1.18·31-s − 3.22·33-s − 0.504·35-s − 0.272·37-s + 0.279·39-s − 1.00·41-s − 1.37·43-s − 1.14·45-s + 0.692·47-s + 0.271·49-s − 0.449·51-s − 1.53·53-s + 0.765·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.199069506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.199069506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 - 2.98T + 7T^{2} \) |
| 11 | \( 1 + 5.67T + 11T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 0.983T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 1.66T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 9.02T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.58T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−9.671020402151551916136843624752, −8.653720395532624664969287551146, −8.196912577529075758923822768303, −7.67070874584651228869675286263, −6.92680621228001270048258224592, −5.13139941065530247102848015866, −4.55801250920256354340095649616, −3.28426194001385779334559992801, −2.66774342767810002611145730859, −1.48725987295006767782210933489,
1.48725987295006767782210933489, 2.66774342767810002611145730859, 3.28426194001385779334559992801, 4.55801250920256354340095649616, 5.13139941065530247102848015866, 6.92680621228001270048258224592, 7.67070874584651228869675286263, 8.196912577529075758923822768303, 8.653720395532624664969287551146, 9.671020402151551916136843624752
