L(s) = 1 | − 1.78·3-s − 0.988·5-s − 3.73·7-s + 0.196·9-s − 5.76·11-s + 4.01·13-s + 1.76·15-s + 6.16·17-s − 4.14·19-s + 6.68·21-s + 1.16·23-s − 4.02·25-s + 5.01·27-s + 5.75·29-s − 1.16·31-s + 10.3·33-s + 3.69·35-s − 9.17·37-s − 7.18·39-s − 7.67·41-s + 3.71·43-s − 0.194·45-s + 5.48·47-s + 6.96·49-s − 11.0·51-s + 1.03·53-s + 5.69·55-s + ⋯ |
L(s) = 1 | − 1.03·3-s − 0.441·5-s − 1.41·7-s + 0.0656·9-s − 1.73·11-s + 1.11·13-s + 0.456·15-s + 1.49·17-s − 0.950·19-s + 1.45·21-s + 0.243·23-s − 0.804·25-s + 0.964·27-s + 1.06·29-s − 0.209·31-s + 1.79·33-s + 0.624·35-s − 1.50·37-s − 1.14·39-s − 1.19·41-s + 0.566·43-s − 0.0290·45-s + 0.800·47-s + 0.995·49-s − 1.54·51-s + 0.141·53-s + 0.767·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 0.988T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 + 3.37T + 83T^{2} \) |
| 89 | \( 1 + 0.772T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.38464947508818958900985830112, −6.62371358786439324705033527267, −6.07017510044687443346269737475, −5.47980853732491313531339764630, −4.92723674119492229468845380480, −3.71247094133442550592382770806, −3.28874554208523382192364355836, −2.33974853884419675898876576035, −0.811853427214395451238250922859, 0,
0.811853427214395451238250922859, 2.33974853884419675898876576035, 3.28874554208523382192364355836, 3.71247094133442550592382770806, 4.92723674119492229468845380480, 5.47980853732491313531339764630, 6.07017510044687443346269737475, 6.62371358786439324705033527267, 7.38464947508818958900985830112